Mark88

No, just sort it out at the appointment.

@SalishSea the OP writes, that "he and his sister were born int he US". That makes them USC by jus soli.
Further the OP writes, that the family returned to Brazil, when the sister was just over a year old. If there weren't any larger breaks, the sister was now more than the required one year in the US.
 
The children are under 18 and born before 11 June 2017. That is why the OP specificly mentioned the pre 2017 bdate, and the this means the "old" rule applies. The old rule (other than the new rule) does not require physcial presence above a certain age, just physical presence prior to the childs birth.
The link I posted above will give you the official embassy information:
https://it.usembassy.gov/u-s-citizen-services/child-family-matters/birth/crba-1/
 
"Old" rule:
For children born on or before June 11, 2017: A child born outside of the United States and out of wedlock to a U.S. citizen mother may acquire U.S. citizenship if the mother was physically present in the United States or one of its outlying possessions for one continuous year (365 days) prior to the birth of the child.
 
"New" rule:
For children born on or after June 12, 2017: A child born outside of the United States out of wedlock to a U.S. citizen mother may acquire U.S. citizenship at birth if the mother was physically present in the United States, or one of its outlying possession, for a period of five years, two of which were after the age of fourteen, prior to the birth of the child.

@SalishSea the OP writes, that "he and his sister were born int he US". That makes them USC by jus soli.
Further the OP writes, that the family returned to Brazil, when the sister was just over a year old. If there weren't any larger breaks, the sister was now more than the required one year in the US.
 
The children are under 18 and born before 11 June 2017. That is why the OP specificly mentioned the pre 2017 bdate, and the this means the "old" rule applies. The old rule (other than the new rule) does not require physcial presence above a certain age, just physical presence prior to the childs birth.
The link I posted above will give you the official embassy information:
https://it.usembassy.gov/u-s-citizen-services/child-family-matters/birth/crba-1/
 
"Old" rule:
For children born on or before June 11, 2017: A child born outside of the United States and out of wedlock to a U.S. citizen mother may acquire U.S. citizenship if the mother was physically present in the United States or one of its outlying possessions for one continuous year (365 days) prior to the birth of the child.
 
"New" rule:
For children born on or after June 12, 2017: A child born outside of the United States out of wedlock to a U.S. citizen mother may acquire U.S. citizenship at birth if the mother was physically present in the United States, or one of its outlying possession, for a period of five years, two of which were after the age of fourteen, prior to the birth of the child.

My wife and I went through a great ordeal getting our child citizenship and in the end it worked out. Since I had numerous questions I couldn't find an answer to in our situation, we decided to write our experience and hope this will help others. Here are the questions and major issues we encountered:
Why and how our Child Report of Birth Abroad (CRBA) was denied at the US Consulate How our child was legally able to enter the US with a baby transportation letter, receives a green card and the process at the airport. Receiving US Citizenship under the Child Citizenship Act of 2000 and a US Passport Applying for a Social Security Number using the new US Passport but without a US birth certificate, and finally Did our child loose her foreign citizenship through be becoming a US Citizen Still deciding to file N-600 with USCIS ---But first, our Background:--

I'm a US Citizen by birth (you can be a naturalized citizen, too. This will not make a difference for the further process) and my wife is a foreign national with a 10 year unconditional green card. We went abroad while she was pregnant to visit her family. While being there, my wife had complications and was hospitalized for a couple days. Because of this, the doctors advised against traveling for her, since the risk was too big of loosing the baby. This is how a 10 day trip turned into 10 months. Since I'm a freelance programmer I had the luxury of conducting my business from abroad and it wasn't a big deal for us. We decided to play it safe and stay. Everything went well and our child was born healthy and without any further complications. We decided to stay a couple of weeks longer so my wife could recover from child birth related injuries and not to put too much stress on the baby with an intercontinental flight.

--Child Report of Birth Abroad and it's denial--
This is when we decided to start a CRBA application with the local US Consulate. We wanted to get US Citizenship for our child and a US Passport to return home. We did start this process two weeks after she was born, since we figured it would take some time to process. But we weren't prepared for this. To be able to transfer your citizenship your child, you have to
be a US Citizen spent at least 5 years in the US, 2 of which have to be after your 14 birthday The child has to be under 18 If the child was born out of wedlock or the US Citizen parent is deceased, or this is for an adopted child there are more restrictions or a total different process. For further information please on the CRBA or the issues mentioned before, please go to your local US Embassy website since our experience is for a biological child born in wedlock.

The CRBA process is also described in numerous forums, so I will keep it short: I had to prove my physical presence in the US. I did spend a lot of time as a child and teenager abroad through my parents work. I brought school records, doctors bills, pay stubs from my parents while employed in the US, business documents of my company, tax returns, my wives green card process which was done in the US(!) - but - all this and much more wasn't enough to satisfy the consular officer. We were dumbstruck by how we were treated and that the amount of evidence didn't accumulate to 5+ years (for them). Our application was denied! If you do this process, be sure to show them at least 10+ years of evidence, since they bump off a certain amount of months for each year.
So what now? My wife and I, legally allowed to live and work in the US, and our 8 week old daughter should stay behind??? I thought I couldn't be the only one in this situation, but after going through google and visa forums I kind of had the feeling we were. But I did find this:

--Greencard for my daughter through my wife--
Ok, so it does happen a lot. People, intentionaly or not, are green card holders and have their children abroad. As described above, in our case this was due to a medical condition. So what now? I did some research and finally contacted Customs And Border Protections (CBP) foreign office at the Embassy. I was lucky to get a quick and competent answer:
If you are the MOTHER and a green card holder AND you have been less than a year abroad (!!! really important !!!) and you are taking your baby on the first trip back to the US (!!! also very important, if out of any reason you - the green card holder - already traveled to the US once without your child, this what comes next won't work !!!) The Embassy/US Consulate issues you a baby transportation letter (it's actually a visa in your baby's foreign passport), also known as "NA-3 (BABY)". We went to the consulate, got the baby transportation letter (took about two hours and can be done on the same day you contact them) and booked a flight for the next week. That was it, and the best part is: Our child AUTOMATICALLY receives a green card upon entering the US. No fees, no wait, no application with USCIS while outside, just automatic. Anyone who dealt with USCIS before knows, things can't be that simple, but in this case its pretty straight forward.

For more information on the Baby Transportation Letter follow this link: http://imgur.com/srhWSt5
To see what a baby transportation letter looks like, click here: http://imgur.com/idQUeWC

Now if you are more than a year abroad, you will basically have to go through the entire green card process for yourself AND your child again. So if you are 360 days outside the US, and you can fly, BUY A TICKET, GET THE TRANSPORTATION LETTER AND GET ON THE PLANE ASAP. Otherwise you might spend weeks, months or even years until you get back and receive another approved green card (+ the $1000's in fees the application will cost you, again). I don't know how DHS determines how long you've been outside for more then a year, but IMHO it's not worth the risk and small children do pretty good on a flight.
Also the transportation letter is confirmation of you being in legal status. So if you're cutting time close, this letter will also be proof that you are eligible to return to the US.

--Processing our child at the airport--
So we landed at the Airport in the US and this is where I didn't find anything on the net or forums what happens next.

You must bring along (so make sure you get this before you fly).
The original foreign birth certificate. - Make sure the father and mother are listed in it. If not, get an "international certificate of birth" at the local civil registry office in the country of birth Translation of the foreign birth certificate. - We had it translated by a certified and public sworn translator. It cost us around $50 but is worth it. This way you can avoid problems of the translation not being legitimate. Oddly USCIS doesn't require a certified translation, BUT CBP, the Social Security Administration (SSA) and Passport Agency at the Department of State (DOS) do! Baby's foreign passport 2 passport size photos (same requirements as for passport, read DOS website http://www.travel.state.gov/passport/pptphotoreq/pptphotoreq_5333.html ). But take a couple more along, just in case (and you'll need them for the US Passport later, too ). We went to an immigration counter and we just showed the immigration officer our passports and the page with the baby transportation letter. He then took us to Secondary Inspection. Don't be worried. This is the area where they process foreigners who don't seem legit to enter the US. But you are there just to wait, nothing more. We did bring other documents (doctors note, ultrasound pics, us tax returns) to support our case, but these were never required by CBP at the airport. We only had to handover the documents and it took about two hours. Afterwards the officer handed us all documents back + a copy of the I-181 which is the proof that the green card was created (this is what you take to the passport agency later, too). CHECK THIS THROUGHLY so that there are no mistake. It will be a real hassle to change anything later + will delay the other parts that follow. They placed a stamp over the NA-3 in the passport and wrote "Permres" in there. That's it. Our child was now a legal permanent resident of the United States and had a (temporary) green card. The real green card comes a couple weeks later in the mail. But it doesn't change that she is a I-551 (green card) holder.

Now comes the real ironic part, that we don't understand why the consulate didn't just grant the CRBA in the first place:

--Automatic Citizenship through the Child Citizenship Act of 2000 (CCA)--

Because our baby entered the US with a green card AND I am a US Citizen the CCA came into place. This means our child was a legal permanent resident in one second BUT became a US Citizen by law in the next second! Puff, Magic, just like that. It took me some time for me to understand the CCA (before we departed back to the US) and even an immigration lawyer I contacted wasn't familiar with this process. USCIS calls this "Derivative Citizenship". Anyway, it works the same way with people who have green cards, naturalize as US Citizens and have children under 18. The moment the parent naturalizes these children with green cards become US Citizens by law. They do not need to apply sepperatly for naturalization. IMHO this law was probably the only good thing George Bush did during his presidency (please no political discussions, this is just my opinion .

So now our child is finally a US Citizen. But of course for her later life she needs proof and who knows if laws change. So the best way to proof your US Citizenship is a US Passport. This is why we applied for it asap.


--Applying for a US Passport under the CCA--
Now, for applying for a US Passport under the CCA you can go to any passport processing location like your local Post Office. If you live in a city like San Francisco or Boston the person processing your application will likely be familiar with the CCA and required documents. If you live in a small town in the square states with hardly any immigrants you will get an evil look and no passport I would recommend going to one of the DOS Passport agencies (especially if you live in one of the cities where these are located). These agencies deal with children, passports and the CCA on a daily basis. This also gave us the advantage of picking up the passport two days later.
At the passport agency we (both parents!) had to appear with our baby. We needed all the documents listed on the link below. The DOS agent was very friendly and knew exactly what to do. They keep all the documents for scanning and filing with the application. Now the DOS recognized our child as a US Citizen and it was official!!! Two days later we picked up the passport with all documents and that was it.

Further information can be found here: http://travel.state.gov/visa/immigrants/types/types_1312.html#7 "How Does the Child Get a Passport Under the Child Citizenship Act?"

--Applying for a Social Security Number with a US Passport and a foreign birth certificate--
I went to my local SSA and brought all documents with me. The agent had no clue what I wanted.
SSAgent: "Only U.S. Citizens can apply for a SSN"
Me: "My child is a U.S. Citizen. You have her US Passport in your hands to prove it."
SSAgent: "Right… But she doesn't have a US birth certificate"
Me: (Long Pause) "And?"
SSAgent: "She needs one, otherwise I can't process the application."
Me: "No she doesn't. Even the http://www.ssa.gov/ssnumber website says she doesn't need one."
SSAgent: "That can't be. I never heard of this. You will need to get her a US birth certificate first"
Me: (Long Pause - since you can't get a US birth certificate if you're not a US Citizen by birth!!!) "Ok. Let's say someone naturalizes and becomes a citizen. They don't have a US birth certificates either. Can't they get a SSN?"
SSAgent: "Right… I will have to check with my supervisor."

Duh! The law which regulates these things has only been in effect for over 12 years!!! Thankfully the supervisor was able to sort this out (after reading the info on the ssa.gov website) and we got the SSN in the mail about 10 days later.

--Did our child loose her foreign citizenship through be becoming a US Citizen?--
Short answer (in our case): NO!
This case varies by each country and has nothing to do with US law. The US can't take away your foreign citizenship, only the country where you (or your child) is originally from can revoke/cancel your foreign citizenship. In our case German law states: If by APPLICATION one shall receive a foreign (in our case, US) citizenship, this leads to loss of German citizenship. Here is the clue: Our daughter didn't apply for US Citizenship, she became a citizen by operation of law ("Derivative Citizenship"). In this case our daughter is a dual citizen. Each country is different so you should check the laws of your country individually.

http://www.germany.info/Vertretung/usa/en/05__Legal/02__Directory__Services/02__Citizenship/Citizenship__Loss.html
http://travel.state.gov/travel/cis_pa_tw/cis/cis_1753.html

-- Certificate of Citizenship through USCIS (N-600) --
Another way for proof of citizenship is a Certificate of Citizenship (COC) through USCIS. I'm still deciding to file this form with USCIS since my child would get a nice official document ending the green card process with USCIS AND this agency "officially recognizing" her citizenship. BUT the reason NOT do it is simple: By law our child is a US Citizen and even the DOS website writes she doesn't need a COC. Her US Passport is proof of her citizenship. AND USCIS will charge you $600+ for just filing the N-600 form + you might have to travel pretty far if a USCIS office won't be able to conduct a short interview in your area (even though USCIS writes you might not even have an interview).
At the same time I applied for a Passport card for her. The cost was only $15 extra and we keep it in a safe place. This way she has backup proof of citizenship if she should ever loses her passport.

http://travel.state.gov/visa/immigrants/types/types_1312.html#6 "Must the Child Get a Certificate of Citizenship?"

--Disclaimer--

This process written above is based on our personal experience. It is in no way legal advice. Feel free for contact me if you have any questions.

I second @iwannaplay54. It will depend alot on the officer, but expect it not to be easy and you needing to provide proof down to weeks spent in the US. You can see our case how hard it was:
 

I second @iwannaplay54. It will depend alot on the officer, but expect it not to be easy and you needing to provide proof down to weeks spent in the US. You can see our case how hard it was:
 

My wife and I went through a great ordeal getting our child citizenship and in the end it worked out. Since I had numerous questions I couldn't find an answer to in our situation, we decided to write our experience and hope this will help others. Here are the questions and major issues we encountered:
Why and how our Child Report of Birth Abroad (CRBA) was denied at the US Consulate How our child was legally able to enter the US with a baby transportation letter, receives a green card and the process at the airport. Receiving US Citizenship under the Child Citizenship Act of 2000 and a US Passport Applying for a Social Security Number using the new US Passport but without a US birth certificate, and finally Did our child loose her foreign citizenship through be becoming a US Citizen Still deciding to file N-600 with USCIS ---But first, our Background:--

I'm a US Citizen by birth (you can be a naturalized citizen, too. This will not make a difference for the further process) and my wife is a foreign national with a 10 year unconditional green card. We went abroad while she was pregnant to visit her family. While being there, my wife had complications and was hospitalized for a couple days. Because of this, the doctors advised against traveling for her, since the risk was too big of loosing the baby. This is how a 10 day trip turned into 10 months. Since I'm a freelance programmer I had the luxury of conducting my business from abroad and it wasn't a big deal for us. We decided to play it safe and stay. Everything went well and our child was born healthy and without any further complications. We decided to stay a couple of weeks longer so my wife could recover from child birth related injuries and not to put too much stress on the baby with an intercontinental flight.

--Child Report of Birth Abroad and it's denial--
This is when we decided to start a CRBA application with the local US Consulate. We wanted to get US Citizenship for our child and a US Passport to return home. We did start this process two weeks after she was born, since we figured it would take some time to process. But we weren't prepared for this. To be able to transfer your citizenship your child, you have to
be a US Citizen spent at least 5 years in the US, 2 of which have to be after your 14 birthday The child has to be under 18 If the child was born out of wedlock or the US Citizen parent is deceased, or this is for an adopted child there are more restrictions or a total different process. For further information please on the CRBA or the issues mentioned before, please go to your local US Embassy website since our experience is for a biological child born in wedlock.

The CRBA process is also described in numerous forums, so I will keep it short: I had to prove my physical presence in the US. I did spend a lot of time as a child and teenager abroad through my parents work. I brought school records, doctors bills, pay stubs from my parents while employed in the US, business documents of my company, tax returns, my wives green card process which was done in the US(!) - but - all this and much more wasn't enough to satisfy the consular officer. We were dumbstruck by how we were treated and that the amount of evidence didn't accumulate to 5+ years (for them). Our application was denied! If you do this process, be sure to show them at least 10+ years of evidence, since they bump off a certain amount of months for each year.
So what now? My wife and I, legally allowed to live and work in the US, and our 8 week old daughter should stay behind??? I thought I couldn't be the only one in this situation, but after going through google and visa forums I kind of had the feeling we were. But I did find this:

--Greencard for my daughter through my wife--
Ok, so it does happen a lot. People, intentionaly or not, are green card holders and have their children abroad. As described above, in our case this was due to a medical condition. So what now? I did some research and finally contacted Customs And Border Protections (CBP) foreign office at the Embassy. I was lucky to get a quick and competent answer:
If you are the MOTHER and a green card holder AND you have been less than a year abroad (!!! really important !!!) and you are taking your baby on the first trip back to the US (!!! also very important, if out of any reason you - the green card holder - already traveled to the US once without your child, this what comes next won't work !!!) The Embassy/US Consulate issues you a baby transportation letter (it's actually a visa in your baby's foreign passport), also known as "NA-3 (BABY)". We went to the consulate, got the baby transportation letter (took about two hours and can be done on the same day you contact them) and booked a flight for the next week. That was it, and the best part is: Our child AUTOMATICALLY receives a green card upon entering the US. No fees, no wait, no application with USCIS while outside, just automatic. Anyone who dealt with USCIS before knows, things can't be that simple, but in this case its pretty straight forward.

For more information on the Baby Transportation Letter follow this link: http://imgur.com/srhWSt5
To see what a baby transportation letter looks like, click here: http://imgur.com/idQUeWC

Now if you are more than a year abroad, you will basically have to go through the entire green card process for yourself AND your child again. So if you are 360 days outside the US, and you can fly, BUY A TICKET, GET THE TRANSPORTATION LETTER AND GET ON THE PLANE ASAP. Otherwise you might spend weeks, months or even years until you get back and receive another approved green card (+ the $1000's in fees the application will cost you, again). I don't know how DHS determines how long you've been outside for more then a year, but IMHO it's not worth the risk and small children do pretty good on a flight.
Also the transportation letter is confirmation of you being in legal status. So if you're cutting time close, this letter will also be proof that you are eligible to return to the US.

--Processing our child at the airport--
So we landed at the Airport in the US and this is where I didn't find anything on the net or forums what happens next.

You must bring along (so make sure you get this before you fly).
The original foreign birth certificate. - Make sure the father and mother are listed in it. If not, get an "international certificate of birth" at the local civil registry office in the country of birth Translation of the foreign birth certificate. - We had it translated by a certified and public sworn translator. It cost us around $50 but is worth it. This way you can avoid problems of the translation not being legitimate. Oddly USCIS doesn't require a certified translation, BUT CBP, the Social Security Administration (SSA) and Passport Agency at the Department of State (DOS) do! Baby's foreign passport 2 passport size photos (same requirements as for passport, read DOS website http://www.travel.state.gov/passport/pptphotoreq/pptphotoreq_5333.html ). But take a couple more along, just in case (and you'll need them for the US Passport later, too ). We went to an immigration counter and we just showed the immigration officer our passports and the page with the baby transportation letter. He then took us to Secondary Inspection. Don't be worried. This is the area where they process foreigners who don't seem legit to enter the US. But you are there just to wait, nothing more. We did bring other documents (doctors note, ultrasound pics, us tax returns) to support our case, but these were never required by CBP at the airport. We only had to handover the documents and it took about two hours. Afterwards the officer handed us all documents back + a copy of the I-181 which is the proof that the green card was created (this is what you take to the passport agency later, too). CHECK THIS THROUGHLY so that there are no mistake. It will be a real hassle to change anything later + will delay the other parts that follow. They placed a stamp over the NA-3 in the passport and wrote "Permres" in there. That's it. Our child was now a legal permanent resident of the United States and had a (temporary) green card. The real green card comes a couple weeks later in the mail. But it doesn't change that she is a I-551 (green card) holder.

Now comes the real ironic part, that we don't understand why the consulate didn't just grant the CRBA in the first place:

--Automatic Citizenship through the Child Citizenship Act of 2000 (CCA)--

Because our baby entered the US with a green card AND I am a US Citizen the CCA came into place. This means our child was a legal permanent resident in one second BUT became a US Citizen by law in the next second! Puff, Magic, just like that. It took me some time for me to understand the CCA (before we departed back to the US) and even an immigration lawyer I contacted wasn't familiar with this process. USCIS calls this "Derivative Citizenship". Anyway, it works the same way with people who have green cards, naturalize as US Citizens and have children under 18. The moment the parent naturalizes these children with green cards become US Citizens by law. They do not need to apply sepperatly for naturalization. IMHO this law was probably the only good thing George Bush did during his presidency (please no political discussions, this is just my opinion .

So now our child is finally a US Citizen. But of course for her later life she needs proof and who knows if laws change. So the best way to proof your US Citizenship is a US Passport. This is why we applied for it asap.


--Applying for a US Passport under the CCA--
Now, for applying for a US Passport under the CCA you can go to any passport processing location like your local Post Office. If you live in a city like San Francisco or Boston the person processing your application will likely be familiar with the CCA and required documents. If you live in a small town in the square states with hardly any immigrants you will get an evil look and no passport I would recommend going to one of the DOS Passport agencies (especially if you live in one of the cities where these are located). These agencies deal with children, passports and the CCA on a daily basis. This also gave us the advantage of picking up the passport two days later.
At the passport agency we (both parents!) had to appear with our baby. We needed all the documents listed on the link below. The DOS agent was very friendly and knew exactly what to do. They keep all the documents for scanning and filing with the application. Now the DOS recognized our child as a US Citizen and it was official!!! Two days later we picked up the passport with all documents and that was it.

Further information can be found here: http://travel.state.gov/visa/immigrants/types/types_1312.html#7 "How Does the Child Get a Passport Under the Child Citizenship Act?"

--Applying for a Social Security Number with a US Passport and a foreign birth certificate--
I went to my local SSA and brought all documents with me. The agent had no clue what I wanted.
SSAgent: "Only U.S. Citizens can apply for a SSN"
Me: "My child is a U.S. Citizen. You have her US Passport in your hands to prove it."
SSAgent: "Right… But she doesn't have a US birth certificate"
Me: (Long Pause) "And?"
SSAgent: "She needs one, otherwise I can't process the application."
Me: "No she doesn't. Even the http://www.ssa.gov/ssnumber website says she doesn't need one."
SSAgent: "That can't be. I never heard of this. You will need to get her a US birth certificate first"
Me: (Long Pause - since you can't get a US birth certificate if you're not a US Citizen by birth!!!) "Ok. Let's say someone naturalizes and becomes a citizen. They don't have a US birth certificates either. Can't they get a SSN?"
SSAgent: "Right… I will have to check with my supervisor."

Duh! The law which regulates these things has only been in effect for over 12 years!!! Thankfully the supervisor was able to sort this out (after reading the info on the ssa.gov website) and we got the SSN in the mail about 10 days later.

--Did our child loose her foreign citizenship through be becoming a US Citizen?--
Short answer (in our case): NO!
This case varies by each country and has nothing to do with US law. The US can't take away your foreign citizenship, only the country where you (or your child) is originally from can revoke/cancel your foreign citizenship. In our case German law states: If by APPLICATION one shall receive a foreign (in our case, US) citizenship, this leads to loss of German citizenship. Here is the clue: Our daughter didn't apply for US Citizenship, she became a citizen by operation of law ("Derivative Citizenship"). In this case our daughter is a dual citizen. Each country is different so you should check the laws of your country individually.

http://www.germany.info/Vertretung/usa/en/05__Legal/02__Directory__Services/02__Citizenship/Citizenship__Loss.html
http://travel.state.gov/travel/cis_pa_tw/cis/cis_1753.html

-- Certificate of Citizenship through USCIS (N-600) --
Another way for proof of citizenship is a Certificate of Citizenship (COC) through USCIS. I'm still deciding to file this form with USCIS since my child would get a nice official document ending the green card process with USCIS AND this agency "officially recognizing" her citizenship. BUT the reason NOT do it is simple: By law our child is a US Citizen and even the DOS website writes she doesn't need a COC. Her US Passport is proof of her citizenship. AND USCIS will charge you $600+ for just filing the N-600 form + you might have to travel pretty far if a USCIS office won't be able to conduct a short interview in your area (even though USCIS writes you might not even have an interview).
At the same time I applied for a Passport card for her. The cost was only $15 extra and we keep it in a safe place. This way she has backup proof of citizenship if she should ever loses her passport.

http://travel.state.gov/visa/immigrants/types/types_1312.html#6 "Must the Child Get a Certificate of Citizenship?"

--Disclaimer--

This process written above is based on our personal experience. It is in no way legal advice. Feel free for contact me if you have any questions.

It will depend on where she was born and where you live:
 
I.e. if you live in Austria and she was born in Germany, they will most likely have you go to the consulate in Munich.
If you live in Portugal and she was born in the UK, then the embassy in Portugal will most likely do the processing.
 
However as @Kor2USA wrote, best thing is to contact them directly.
 
Processing time is pretty much the same. Usually within the hour.

My wife and I went through a great ordeal getting our child citizenship and in the end it worked out. Since I had numerous questions I couldn't find an answer to in our situation, we decided to write our experience and hope this will help others. Here are the questions and major issues we encountered:
Why and how our Child Report of Birth Abroad (CRBA) was denied at the US Consulate How our child was legally able to enter the US with a baby transportation letter, receives a green card and the process at the airport. Receiving US Citizenship under the Child Citizenship Act of 2000 and a US Passport Applying for a Social Security Number using the new US Passport but without a US birth certificate, and finally Did our child loose her foreign citizenship through be becoming a US Citizen Still deciding to file N-600 with USCIS ---But first, our Background:--

I'm a US Citizen by birth (you can be a naturalized citizen, too. This will not make a difference for the further process) and my wife is a foreign national with a 10 year unconditional green card. We went abroad while she was pregnant to visit her family. While being there, my wife had complications and was hospitalized for a couple days. Because of this, the doctors advised against traveling for her, since the risk was too big of loosing the baby. This is how a 10 day trip turned into 10 months. Since I'm a freelance programmer I had the luxury of conducting my business from abroad and it wasn't a big deal for us. We decided to play it safe and stay. Everything went well and our child was born healthy and without any further complications. We decided to stay a couple of weeks longer so my wife could recover from child birth related injuries and not to put too much stress on the baby with an intercontinental flight.

--Child Report of Birth Abroad and it's denial--
This is when we decided to start a CRBA application with the local US Consulate. We wanted to get US Citizenship for our child and a US Passport to return home. We did start this process two weeks after she was born, since we figured it would take some time to process. But we weren't prepared for this. To be able to transfer your citizenship your child, you have to
be a US Citizen spent at least 5 years in the US, 2 of which have to be after your 14 birthday The child has to be under 18 If the child was born out of wedlock or the US Citizen parent is deceased, or this is for an adopted child there are more restrictions or a total different process. For further information please on the CRBA or the issues mentioned before, please go to your local US Embassy website since our experience is for a biological child born in wedlock.

The CRBA process is also described in numerous forums, so I will keep it short: I had to prove my physical presence in the US. I did spend a lot of time as a child and teenager abroad through my parents work. I brought school records, doctors bills, pay stubs from my parents while employed in the US, business documents of my company, tax returns, my wives green card process which was done in the US(!) - but - all this and much more wasn't enough to satisfy the consular officer. We were dumbstruck by how we were treated and that the amount of evidence didn't accumulate to 5+ years (for them). Our application was denied! If you do this process, be sure to show them at least 10+ years of evidence, since they bump off a certain amount of months for each year.
So what now? My wife and I, legally allowed to live and work in the US, and our 8 week old daughter should stay behind??? I thought I couldn't be the only one in this situation, but after going through google and visa forums I kind of had the feeling we were. But I did find this:

--Greencard for my daughter through my wife--
Ok, so it does happen a lot. People, intentionaly or not, are green card holders and have their children abroad. As described above, in our case this was due to a medical condition. So what now? I did some research and finally contacted Customs And Border Protections (CBP) foreign office at the Embassy. I was lucky to get a quick and competent answer:
If you are the MOTHER and a green card holder AND you have been less than a year abroad (!!! really important !!!) and you are taking your baby on the first trip back to the US (!!! also very important, if out of any reason you - the green card holder - already traveled to the US once without your child, this what comes next won't work !!!) The Embassy/US Consulate issues you a baby transportation letter (it's actually a visa in your baby's foreign passport), also known as "NA-3 (BABY)". We went to the consulate, got the baby transportation letter (took about two hours and can be done on the same day you contact them) and booked a flight for the next week. That was it, and the best part is: Our child AUTOMATICALLY receives a green card upon entering the US. No fees, no wait, no application with USCIS while outside, just automatic. Anyone who dealt with USCIS before knows, things can't be that simple, but in this case its pretty straight forward.

For more information on the Baby Transportation Letter follow this link: http://imgur.com/srhWSt5
To see what a baby transportation letter looks like, click here: http://imgur.com/idQUeWC

Now if you are more than a year abroad, you will basically have to go through the entire green card process for yourself AND your child again. So if you are 360 days outside the US, and you can fly, BUY A TICKET, GET THE TRANSPORTATION LETTER AND GET ON THE PLANE ASAP. Otherwise you might spend weeks, months or even years until you get back and receive another approved green card (+ the $1000's in fees the application will cost you, again). I don't know how DHS determines how long you've been outside for more then a year, but IMHO it's not worth the risk and small children do pretty good on a flight.
Also the transportation letter is confirmation of you being in legal status. So if you're cutting time close, this letter will also be proof that you are eligible to return to the US.

--Processing our child at the airport--
So we landed at the Airport in the US and this is where I didn't find anything on the net or forums what happens next.

You must bring along (so make sure you get this before you fly).
The original foreign birth certificate. - Make sure the father and mother are listed in it. If not, get an "international certificate of birth" at the local civil registry office in the country of birth Translation of the foreign birth certificate. - We had it translated by a certified and public sworn translator. It cost us around $50 but is worth it. This way you can avoid problems of the translation not being legitimate. Oddly USCIS doesn't require a certified translation, BUT CBP, the Social Security Administration (SSA) and Passport Agency at the Department of State (DOS) do! Baby's foreign passport 2 passport size photos (same requirements as for passport, read DOS website http://www.travel.state.gov/passport/pptphotoreq/pptphotoreq_5333.html ). But take a couple more along, just in case (and you'll need them for the US Passport later, too ). We went to an immigration counter and we just showed the immigration officer our passports and the page with the baby transportation letter. He then took us to Secondary Inspection. Don't be worried. This is the area where they process foreigners who don't seem legit to enter the US. But you are there just to wait, nothing more. We did bring other documents (doctors note, ultrasound pics, us tax returns) to support our case, but these were never required by CBP at the airport. We only had to handover the documents and it took about two hours. Afterwards the officer handed us all documents back + a copy of the I-181 which is the proof that the green card was created (this is what you take to the passport agency later, too). CHECK THIS THROUGHLY so that there are no mistake. It will be a real hassle to change anything later + will delay the other parts that follow. They placed a stamp over the NA-3 in the passport and wrote "Permres" in there. That's it. Our child was now a legal permanent resident of the United States and had a (temporary) green card. The real green card comes a couple weeks later in the mail. But it doesn't change that she is a I-551 (green card) holder.

Now comes the real ironic part, that we don't understand why the consulate didn't just grant the CRBA in the first place:

--Automatic Citizenship through the Child Citizenship Act of 2000 (CCA)--

Because our baby entered the US with a green card AND I am a US Citizen the CCA came into place. This means our child was a legal permanent resident in one second BUT became a US Citizen by law in the next second! Puff, Magic, just like that. It took me some time for me to understand the CCA (before we departed back to the US) and even an immigration lawyer I contacted wasn't familiar with this process. USCIS calls this "Derivative Citizenship". Anyway, it works the same way with people who have green cards, naturalize as US Citizens and have children under 18. The moment the parent naturalizes these children with green cards become US Citizens by law. They do not need to apply sepperatly for naturalization. IMHO this law was probably the only good thing George Bush did during his presidency (please no political discussions, this is just my opinion .

So now our child is finally a US Citizen. But of course for her later life she needs proof and who knows if laws change. So the best way to proof your US Citizenship is a US Passport. This is why we applied for it asap.


--Applying for a US Passport under the CCA--
Now, for applying for a US Passport under the CCA you can go to any passport processing location like your local Post Office. If you live in a city like San Francisco or Boston the person processing your application will likely be familiar with the CCA and required documents. If you live in a small town in the square states with hardly any immigrants you will get an evil look and no passport I would recommend going to one of the DOS Passport agencies (especially if you live in one of the cities where these are located). These agencies deal with children, passports and the CCA on a daily basis. This also gave us the advantage of picking up the passport two days later.
At the passport agency we (both parents!) had to appear with our baby. We needed all the documents listed on the link below. The DOS agent was very friendly and knew exactly what to do. They keep all the documents for scanning and filing with the application. Now the DOS recognized our child as a US Citizen and it was official!!! Two days later we picked up the passport with all documents and that was it.

Further information can be found here: http://travel.state.gov/visa/immigrants/types/types_1312.html#7 "How Does the Child Get a Passport Under the Child Citizenship Act?"

--Applying for a Social Security Number with a US Passport and a foreign birth certificate--
I went to my local SSA and brought all documents with me. The agent had no clue what I wanted.
SSAgent: "Only U.S. Citizens can apply for a SSN"
Me: "My child is a U.S. Citizen. You have her US Passport in your hands to prove it."
SSAgent: "Right… But she doesn't have a US birth certificate"
Me: (Long Pause) "And?"
SSAgent: "She needs one, otherwise I can't process the application."
Me: "No she doesn't. Even the http://www.ssa.gov/ssnumber website says she doesn't need one."
SSAgent: "That can't be. I never heard of this. You will need to get her a US birth certificate first"
Me: (Long Pause - since you can't get a US birth certificate if you're not a US Citizen by birth!!!) "Ok. Let's say someone naturalizes and becomes a citizen. They don't have a US birth certificates either. Can't they get a SSN?"
SSAgent: "Right… I will have to check with my supervisor."

Duh! The law which regulates these things has only been in effect for over 12 years!!! Thankfully the supervisor was able to sort this out (after reading the info on the ssa.gov website) and we got the SSN in the mail about 10 days later.

--Did our child loose her foreign citizenship through be becoming a US Citizen?--
Short answer (in our case): NO!
This case varies by each country and has nothing to do with US law. The US can't take away your foreign citizenship, only the country where you (or your child) is originally from can revoke/cancel your foreign citizenship. In our case German law states: If by APPLICATION one shall receive a foreign (in our case, US) citizenship, this leads to loss of German citizenship. Here is the clue: Our daughter didn't apply for US Citizenship, she became a citizen by operation of law ("Derivative Citizenship"). In this case our daughter is a dual citizen. Each country is different so you should check the laws of your country individually.

http://www.germany.info/Vertretung/usa/en/05__Legal/02__Directory__Services/02__Citizenship/Citizenship__Loss.html
http://travel.state.gov/travel/cis_pa_tw/cis/cis_1753.html

-- Certificate of Citizenship through USCIS (N-600) --
Another way for proof of citizenship is a Certificate of Citizenship (COC) through USCIS. I'm still deciding to file this form with USCIS since my child would get a nice official document ending the green card process with USCIS AND this agency "officially recognizing" her citizenship. BUT the reason NOT do it is simple: By law our child is a US Citizen and even the DOS website writes she doesn't need a COC. Her US Passport is proof of her citizenship. AND USCIS will charge you $600+ for just filing the N-600 form + you might have to travel pretty far if a USCIS office won't be able to conduct a short interview in your area (even though USCIS writes you might not even have an interview).
At the same time I applied for a Passport card for her. The cost was only $15 extra and we keep it in a safe place. This way she has backup proof of citizenship if she should ever loses her passport.

http://travel.state.gov/visa/immigrants/types/types_1312.html#6 "Must the Child Get a Certificate of Citizenship?"

--Disclaimer--

This process written above is based on our personal experience. It is in no way legal advice. Feel free for contact me if you have any questions.

My wife and I went through a great ordeal getting our child citizenship and in the end it worked out. Since I had numerous questions I couldn't find an answer to in our situation, we decided to write our experience and hope this will help others. Here are the questions and major issues we encountered:
Why and how our Child Report of Birth Abroad (CRBA) was denied at the US Consulate How our child was legally able to enter the US with a baby transportation letter, receives a green card and the process at the airport. Receiving US Citizenship under the Child Citizenship Act of 2000 and a US Passport Applying for a Social Security Number using the new US Passport but without a US birth certificate, and finally Did our child loose her foreign citizenship through be becoming a US Citizen Still deciding to file N-600 with USCIS ---But first, our Background:--

I'm a US Citizen by birth (you can be a naturalized citizen, too. This will not make a difference for the further process) and my wife is a foreign national with a 10 year unconditional green card. We went abroad while she was pregnant to visit her family. While being there, my wife had complications and was hospitalized for a couple days. Because of this, the doctors advised against traveling for her, since the risk was too big of loosing the baby. This is how a 10 day trip turned into 10 months. Since I'm a freelance programmer I had the luxury of conducting my business from abroad and it wasn't a big deal for us. We decided to play it safe and stay. Everything went well and our child was born healthy and without any further complications. We decided to stay a couple of weeks longer so my wife could recover from child birth related injuries and not to put too much stress on the baby with an intercontinental flight.

--Child Report of Birth Abroad and it's denial--
This is when we decided to start a CRBA application with the local US Consulate. We wanted to get US Citizenship for our child and a US Passport to return home. We did start this process two weeks after she was born, since we figured it would take some time to process. But we weren't prepared for this. To be able to transfer your citizenship your child, you have to
be a US Citizen spent at least 5 years in the US, 2 of which have to be after your 14 birthday The child has to be under 18 If the child was born out of wedlock or the US Citizen parent is deceased, or this is for an adopted child there are more restrictions or a total different process. For further information please on the CRBA or the issues mentioned before, please go to your local US Embassy website since our experience is for a biological child born in wedlock.

The CRBA process is also described in numerous forums, so I will keep it short: I had to prove my physical presence in the US. I did spend a lot of time as a child and teenager abroad through my parents work. I brought school records, doctors bills, pay stubs from my parents while employed in the US, business documents of my company, tax returns, my wives green card process which was done in the US(!) - but - all this and much more wasn't enough to satisfy the consular officer. We were dumbstruck by how we were treated and that the amount of evidence didn't accumulate to 5+ years (for them). Our application was denied! If you do this process, be sure to show them at least 10+ years of evidence, since they bump off a certain amount of months for each year.
So what now? My wife and I, legally allowed to live and work in the US, and our 8 week old daughter should stay behind??? I thought I couldn't be the only one in this situation, but after going through google and visa forums I kind of had the feeling we were. But I did find this:

--Greencard for my daughter through my wife--
Ok, so it does happen a lot. People, intentionaly or not, are green card holders and have their children abroad. As described above, in our case this was due to a medical condition. So what now? I did some research and finally contacted Customs And Border Protections (CBP) foreign office at the Embassy. I was lucky to get a quick and competent answer:
If you are the MOTHER and a green card holder AND you have been less than a year abroad (!!! really important !!!) and you are taking your baby on the first trip back to the US (!!! also very important, if out of any reason you - the green card holder - already traveled to the US once without your child, this what comes next won't work !!!) The Embassy/US Consulate issues you a baby transportation letter (it's actually a visa in your baby's foreign passport), also known as "NA-3 (BABY)". We went to the consulate, got the baby transportation letter (took about two hours and can be done on the same day you contact them) and booked a flight for the next week. That was it, and the best part is: Our child AUTOMATICALLY receives a green card upon entering the US. No fees, no wait, no application with USCIS while outside, just automatic. Anyone who dealt with USCIS before knows, things can't be that simple, but in this case its pretty straight forward.

For more information on the Baby Transportation Letter follow this link: http://imgur.com/srhWSt5
To see what a baby transportation letter looks like, click here: http://imgur.com/idQUeWC

Now if you are more than a year abroad, you will basically have to go through the entire green card process for yourself AND your child again. So if you are 360 days outside the US, and you can fly, BUY A TICKET, GET THE TRANSPORTATION LETTER AND GET ON THE PLANE ASAP. Otherwise you might spend weeks, months or even years until you get back and receive another approved green card (+ the $1000's in fees the application will cost you, again). I don't know how DHS determines how long you've been outside for more then a year, but IMHO it's not worth the risk and small children do pretty good on a flight.
Also the transportation letter is confirmation of you being in legal status. So if you're cutting time close, this letter will also be proof that you are eligible to return to the US.

--Processing our child at the airport--
So we landed at the Airport in the US and this is where I didn't find anything on the net or forums what happens next.

You must bring along (so make sure you get this before you fly).
The original foreign birth certificate. - Make sure the father and mother are listed in it. If not, get an "international certificate of birth" at the local civil registry office in the country of birth Translation of the foreign birth certificate. - We had it translated by a certified and public sworn translator. It cost us around $50 but is worth it. This way you can avoid problems of the translation not being legitimate. Oddly USCIS doesn't require a certified translation, BUT CBP, the Social Security Administration (SSA) and Passport Agency at the Department of State (DOS) do! Baby's foreign passport 2 passport size photos (same requirements as for passport, read DOS website http://www.travel.state.gov/passport/pptphotoreq/pptphotoreq_5333.html ). But take a couple more along, just in case (and you'll need them for the US Passport later, too ). We went to an immigration counter and we just showed the immigration officer our passports and the page with the baby transportation letter. He then took us to Secondary Inspection. Don't be worried. This is the area where they process foreigners who don't seem legit to enter the US. But you are there just to wait, nothing more. We did bring other documents (doctors note, ultrasound pics, us tax returns) to support our case, but these were never required by CBP at the airport. We only had to handover the documents and it took about two hours. Afterwards the officer handed us all documents back + a copy of the I-181 which is the proof that the green card was created (this is what you take to the passport agency later, too). CHECK THIS THROUGHLY so that there are no mistake. It will be a real hassle to change anything later + will delay the other parts that follow. They placed a stamp over the NA-3 in the passport and wrote "Permres" in there. That's it. Our child was now a legal permanent resident of the United States and had a (temporary) green card. The real green card comes a couple weeks later in the mail. But it doesn't change that she is a I-551 (green card) holder.

Now comes the real ironic part, that we don't understand why the consulate didn't just grant the CRBA in the first place:

--Automatic Citizenship through the Child Citizenship Act of 2000 (CCA)--

Because our baby entered the US with a green card AND I am a US Citizen the CCA came into place. This means our child was a legal permanent resident in one second BUT became a US Citizen by law in the next second! Puff, Magic, just like that. It took me some time for me to understand the CCA (before we departed back to the US) and even an immigration lawyer I contacted wasn't familiar with this process. USCIS calls this "Derivative Citizenship". Anyway, it works the same way with people who have green cards, naturalize as US Citizens and have children under 18. The moment the parent naturalizes these children with green cards become US Citizens by law. They do not need to apply sepperatly for naturalization. IMHO this law was probably the only good thing George Bush did during his presidency (please no political discussions, this is just my opinion .

So now our child is finally a US Citizen. But of course for her later life she needs proof and who knows if laws change. So the best way to proof your US Citizenship is a US Passport. This is why we applied for it asap.


--Applying for a US Passport under the CCA--
Now, for applying for a US Passport under the CCA you can go to any passport processing location like your local Post Office. If you live in a city like San Francisco or Boston the person processing your application will likely be familiar with the CCA and required documents. If you live in a small town in the square states with hardly any immigrants you will get an evil look and no passport I would recommend going to one of the DOS Passport agencies (especially if you live in one of the cities where these are located). These agencies deal with children, passports and the CCA on a daily basis. This also gave us the advantage of picking up the passport two days later.
At the passport agency we (both parents!) had to appear with our baby. We needed all the documents listed on the link below. The DOS agent was very friendly and knew exactly what to do. They keep all the documents for scanning and filing with the application. Now the DOS recognized our child as a US Citizen and it was official!!! Two days later we picked up the passport with all documents and that was it.

Further information can be found here: http://travel.state.gov/visa/immigrants/types/types_1312.html#7 "How Does the Child Get a Passport Under the Child Citizenship Act?"

--Applying for a Social Security Number with a US Passport and a foreign birth certificate--
I went to my local SSA and brought all documents with me. The agent had no clue what I wanted.
SSAgent: "Only U.S. Citizens can apply for a SSN"
Me: "My child is a U.S. Citizen. You have her US Passport in your hands to prove it."
SSAgent: "Right… But she doesn't have a US birth certificate"
Me: (Long Pause) "And?"
SSAgent: "She needs one, otherwise I can't process the application."
Me: "No she doesn't. Even the http://www.ssa.gov/ssnumber website says she doesn't need one."
SSAgent: "That can't be. I never heard of this. You will need to get her a US birth certificate first"
Me: (Long Pause - since you can't get a US birth certificate if you're not a US Citizen by birth!!!) "Ok. Let's say someone naturalizes and becomes a citizen. They don't have a US birth certificates either. Can't they get a SSN?"
SSAgent: "Right… I will have to check with my supervisor."

Duh! The law which regulates these things has only been in effect for over 12 years!!! Thankfully the supervisor was able to sort this out (after reading the info on the ssa.gov website) and we got the SSN in the mail about 10 days later.

--Did our child loose her foreign citizenship through be becoming a US Citizen?--
Short answer (in our case): NO!
This case varies by each country and has nothing to do with US law. The US can't take away your foreign citizenship, only the country where you (or your child) is originally from can revoke/cancel your foreign citizenship. In our case German law states: If by APPLICATION one shall receive a foreign (in our case, US) citizenship, this leads to loss of German citizenship. Here is the clue: Our daughter didn't apply for US Citizenship, she became a citizen by operation of law ("Derivative Citizenship"). In this case our daughter is a dual citizen. Each country is different so you should check the laws of your country individually.

http://www.germany.info/Vertretung/usa/en/05__Legal/02__Directory__Services/02__Citizenship/Citizenship__Loss.html
http://travel.state.gov/travel/cis_pa_tw/cis/cis_1753.html

-- Certificate of Citizenship through USCIS (N-600) --
Another way for proof of citizenship is a Certificate of Citizenship (COC) through USCIS. I'm still deciding to file this form with USCIS since my child would get a nice official document ending the green card process with USCIS AND this agency "officially recognizing" her citizenship. BUT the reason NOT do it is simple: By law our child is a US Citizen and even the DOS website writes she doesn't need a COC. Her US Passport is proof of her citizenship. AND USCIS will charge you $600+ for just filing the N-600 form + you might have to travel pretty far if a USCIS office won't be able to conduct a short interview in your area (even though USCIS writes you might not even have an interview).
At the same time I applied for a Passport card for her. The cost was only $15 extra and we keep it in a safe place. This way she has backup proof of citizenship if she should ever loses her passport.

http://travel.state.gov/visa/immigrants/types/types_1312.html#6 "Must the Child Get a Certificate of Citizenship?"

--Disclaimer--

This process written above is based on our personal experience. It is in no way legal advice. Feel free for contact me if you have any questions.

Your question entangeles a lot of immigration and citizenship procedures and can not be answered in one sentence. You basicly have 3 options:
 
1) CRBA (which only can be done outside the US)
This is also the easiest way and your baby will get a US passport. The consulate can even issue an emergency passport on the same day. Call them, contact them by email, they should somehow get you an earlier appointment. Also the issuance of a CRBA will get your child a document that can substitute as a US birth certificate.
 
2) Travel to the US on a Canadian Passport
Your child could be denied entry to the US, since you are entering on an immigrant visa and your child lacks proper documentation. This can lead then to a major issue for both of you, since you will probaly not enter the US without your child and you would jeopardize your K1 visa. If you are able to enter the US eventhough concerning the above isse, you can just go to together with the father to a passport agency and apply for a US Passport there. It is the same process as a CRBA (i.e. the father will have to fulfill the physical presence requirements), but only a passport and no CRBA will be issued.
 
3) File a seperate petiton for your child
Not going to work, since your child has a USC father and this way no visa will be issued by the consulate.
 
Considering all this, just GET A CRBA.

Whatever name you give your child - as long within the Standesamtsrecht - it is your decision. But you do see the issues it can cause. 
 
Why does the US want your husband to change his name? Is it too long, too? 
 
And that a passport aplied from abroad can only have 20 characters does not seem to make sense to me: My passport (including my very first one in my life and the most recent one + my children) were applied from abroad (the US consulate in Germany to be exact) and my name and those of my children have 20+ characters too (under 33 characters though). 
 
And the "von" is not illegal per se, but since Germany abolished royalty, there is just no King or Emperor around who can give you a "von" title

A traditional German name with 45 characters? Unless it ends with "von Bayern" or "zu Hohenzollern" which name did he give the child? 
 
And to your problem, do you already have the CRBA?

My wife and I went through a great ordeal getting our child citizenship and in the end it worked out. Since I had numerous questions I couldn't find an answer to in our situation, we decided to write our experience and hope this will help others. Here are the questions and major issues we encountered:
Why and how our Child Report of Birth Abroad (CRBA) was denied at the US Consulate How our child was legally able to enter the US with a baby transportation letter, receives a green card and the process at the airport. Receiving US Citizenship under the Child Citizenship Act of 2000 and a US Passport Applying for a Social Security Number using the new US Passport but without a US birth certificate, and finally Did our child loose her foreign citizenship through be becoming a US Citizen Still deciding to file N-600 with USCIS ---But first, our Background:--

I'm a US Citizen by birth (you can be a naturalized citizen, too. This will not make a difference for the further process) and my wife is a foreign national with a 10 year unconditional green card. We went abroad while she was pregnant to visit her family. While being there, my wife had complications and was hospitalized for a couple days. Because of this, the doctors advised against traveling for her, since the risk was too big of loosing the baby. This is how a 10 day trip turned into 10 months. Since I'm a freelance programmer I had the luxury of conducting my business from abroad and it wasn't a big deal for us. We decided to play it safe and stay. Everything went well and our child was born healthy and without any further complications. We decided to stay a couple of weeks longer so my wife could recover from child birth related injuries and not to put too much stress on the baby with an intercontinental flight.

--Child Report of Birth Abroad and it's denial--
This is when we decided to start a CRBA application with the local US Consulate. We wanted to get US Citizenship for our child and a US Passport to return home. We did start this process two weeks after she was born, since we figured it would take some time to process. But we weren't prepared for this. To be able to transfer your citizenship your child, you have to
be a US Citizen spent at least 5 years in the US, 2 of which have to be after your 14 birthday The child has to be under 18 If the child was born out of wedlock or the US Citizen parent is deceased, or this is for an adopted child there are more restrictions or a total different process. For further information please on the CRBA or the issues mentioned before, please go to your local US Embassy website since our experience is for a biological child born in wedlock.

The CRBA process is also described in numerous forums, so I will keep it short: I had to prove my physical presence in the US. I did spend a lot of time as a child and teenager abroad through my parents work. I brought school records, doctors bills, pay stubs from my parents while employed in the US, business documents of my company, tax returns, my wives green card process which was done in the US(!) - but - all this and much more wasn't enough to satisfy the consular officer. We were dumbstruck by how we were treated and that the amount of evidence didn't accumulate to 5+ years (for them). Our application was denied! If you do this process, be sure to show them at least 10+ years of evidence, since they bump off a certain amount of months for each year.
So what now? My wife and I, legally allowed to live and work in the US, and our 8 week old daughter should stay behind??? I thought I couldn't be the only one in this situation, but after going through google and visa forums I kind of had the feeling we were. But I did find this:

--Greencard for my daughter through my wife--
Ok, so it does happen a lot. People, intentionaly or not, are green card holders and have their children abroad. As described above, in our case this was due to a medical condition. So what now? I did some research and finally contacted Customs And Border Protections (CBP) foreign office at the Embassy. I was lucky to get a quick and competent answer:
If you are the MOTHER and a green card holder AND you have been less than a year abroad (!!! really important !!!) and you are taking your baby on the first trip back to the US (!!! also very important, if out of any reason you - the green card holder - already traveled to the US once without your child, this what comes next won't work !!!) The Embassy/US Consulate issues you a baby transportation letter (it's actually a visa in your baby's foreign passport), also known as "NA-3 (BABY)". We went to the consulate, got the baby transportation letter (took about two hours and can be done on the same day you contact them) and booked a flight for the next week. That was it, and the best part is: Our child AUTOMATICALLY receives a green card upon entering the US. No fees, no wait, no application with USCIS while outside, just automatic. Anyone who dealt with USCIS before knows, things can't be that simple, but in this case its pretty straight forward.

For more information on the Baby Transportation Letter follow this link: http://imgur.com/srhWSt5
To see what a baby transportation letter looks like, click here: http://imgur.com/idQUeWC

Now if you are more than a year abroad, you will basically have to go through the entire green card process for yourself AND your child again. So if you are 360 days outside the US, and you can fly, BUY A TICKET, GET THE TRANSPORTATION LETTER AND GET ON THE PLANE ASAP. Otherwise you might spend weeks, months or even years until you get back and receive another approved green card (+ the $1000's in fees the application will cost you, again). I don't know how DHS determines how long you've been outside for more then a year, but IMHO it's not worth the risk and small children do pretty good on a flight.
Also the transportation letter is confirmation of you being in legal status. So if you're cutting time close, this letter will also be proof that you are eligible to return to the US.

--Processing our child at the airport--
So we landed at the Airport in the US and this is where I didn't find anything on the net or forums what happens next.

You must bring along (so make sure you get this before you fly).
The original foreign birth certificate. - Make sure the father and mother are listed in it. If not, get an "international certificate of birth" at the local civil registry office in the country of birth Translation of the foreign birth certificate. - We had it translated by a certified and public sworn translator. It cost us around $50 but is worth it. This way you can avoid problems of the translation not being legitimate. Oddly USCIS doesn't require a certified translation, BUT CBP, the Social Security Administration (SSA) and Passport Agency at the Department of State (DOS) do! Baby's foreign passport 2 passport size photos (same requirements as for passport, read DOS website http://www.travel.state.gov/passport/pptphotoreq/pptphotoreq_5333.html ). But take a couple more along, just in case (and you'll need them for the US Passport later, too ). We went to an immigration counter and we just showed the immigration officer our passports and the page with the baby transportation letter. He then took us to Secondary Inspection. Don't be worried. This is the area where they process foreigners who don't seem legit to enter the US. But you are there just to wait, nothing more. We did bring other documents (doctors note, ultrasound pics, us tax returns) to support our case, but these were never required by CBP at the airport. We only had to handover the documents and it took about two hours. Afterwards the officer handed us all documents back + a copy of the I-181 which is the proof that the green card was created (this is what you take to the passport agency later, too). CHECK THIS THROUGHLY so that there are no mistake. It will be a real hassle to change anything later + will delay the other parts that follow. They placed a stamp over the NA-3 in the passport and wrote "Permres" in there. That's it. Our child was now a legal permanent resident of the United States and had a (temporary) green card. The real green card comes a couple weeks later in the mail. But it doesn't change that she is a I-551 (green card) holder.

Now comes the real ironic part, that we don't understand why the consulate didn't just grant the CRBA in the first place:

--Automatic Citizenship through the Child Citizenship Act of 2000 (CCA)--

Because our baby entered the US with a green card AND I am a US Citizen the CCA came into place. This means our child was a legal permanent resident in one second BUT became a US Citizen by law in the next second! Puff, Magic, just like that. It took me some time for me to understand the CCA (before we departed back to the US) and even an immigration lawyer I contacted wasn't familiar with this process. USCIS calls this "Derivative Citizenship". Anyway, it works the same way with people who have green cards, naturalize as US Citizens and have children under 18. The moment the parent naturalizes these children with green cards become US Citizens by law. They do not need to apply sepperatly for naturalization. IMHO this law was probably the only good thing George Bush did during his presidency (please no political discussions, this is just my opinion .

So now our child is finally a US Citizen. But of course for her later life she needs proof and who knows if laws change. So the best way to proof your US Citizenship is a US Passport. This is why we applied for it asap.


--Applying for a US Passport under the CCA--
Now, for applying for a US Passport under the CCA you can go to any passport processing location like your local Post Office. If you live in a city like San Francisco or Boston the person processing your application will likely be familiar with the CCA and required documents. If you live in a small town in the square states with hardly any immigrants you will get an evil look and no passport I would recommend going to one of the DOS Passport agencies (especially if you live in one of the cities where these are located). These agencies deal with children, passports and the CCA on a daily basis. This also gave us the advantage of picking up the passport two days later.
At the passport agency we (both parents!) had to appear with our baby. We needed all the documents listed on the link below. The DOS agent was very friendly and knew exactly what to do. They keep all the documents for scanning and filing with the application. Now the DOS recognized our child as a US Citizen and it was official!!! Two days later we picked up the passport with all documents and that was it.

Further information can be found here: http://travel.state.gov/visa/immigrants/types/types_1312.html#7 "How Does the Child Get a Passport Under the Child Citizenship Act?"

--Applying for a Social Security Number with a US Passport and a foreign birth certificate--
I went to my local SSA and brought all documents with me. The agent had no clue what I wanted.
SSAgent: "Only U.S. Citizens can apply for a SSN"
Me: "My child is a U.S. Citizen. You have her US Passport in your hands to prove it."
SSAgent: "Right… But she doesn't have a US birth certificate"
Me: (Long Pause) "And?"
SSAgent: "She needs one, otherwise I can't process the application."
Me: "No she doesn't. Even the http://www.ssa.gov/ssnumber website says she doesn't need one."
SSAgent: "That can't be. I never heard of this. You will need to get her a US birth certificate first"
Me: (Long Pause - since you can't get a US birth certificate if you're not a US Citizen by birth!!!) "Ok. Let's say someone naturalizes and becomes a citizen. They don't have a US birth certificates either. Can't they get a SSN?"
SSAgent: "Right… I will have to check with my supervisor."

Duh! The law which regulates these things has only been in effect for over 12 years!!! Thankfully the supervisor was able to sort this out (after reading the info on the ssa.gov website) and we got the SSN in the mail about 10 days later.

--Did our child loose her foreign citizenship through be becoming a US Citizen?--
Short answer (in our case): NO!
This case varies by each country and has nothing to do with US law. The US can't take away your foreign citizenship, only the country where you (or your child) is originally from can revoke/cancel your foreign citizenship. In our case German law states: If by APPLICATION one shall receive a foreign (in our case, US) citizenship, this leads to loss of German citizenship. Here is the clue: Our daughter didn't apply for US Citizenship, she became a citizen by operation of law ("Derivative Citizenship"). In this case our daughter is a dual citizen. Each country is different so you should check the laws of your country individually.

http://www.germany.info/Vertretung/usa/en/05__Legal/02__Directory__Services/02__Citizenship/Citizenship__Loss.html
http://travel.state.gov/travel/cis_pa_tw/cis/cis_1753.html

-- Certificate of Citizenship through USCIS (N-600) --
Another way for proof of citizenship is a Certificate of Citizenship (COC) through USCIS. I'm still deciding to file this form with USCIS since my child would get a nice official document ending the green card process with USCIS AND this agency "officially recognizing" her citizenship. BUT the reason NOT do it is simple: By law our child is a US Citizen and even the DOS website writes she doesn't need a COC. Her US Passport is proof of her citizenship. AND USCIS will charge you $600+ for just filing the N-600 form + you might have to travel pretty far if a USCIS office won't be able to conduct a short interview in your area (even though USCIS writes you might not even have an interview).
At the same time I applied for a Passport card for her. The cost was only $15 extra and we keep it in a safe place. This way she has backup proof of citizenship if she should ever loses her passport.

http://travel.state.gov/visa/immigrants/types/types_1312.html#6 "Must the Child Get a Certificate of Citizenship?"

--Disclaimer--

This process written above is based on our personal experience. It is in no way legal advice. Feel free for contact me if you have any questions.

I agree, they were lucky in so many ways:
a) No valid ESTA -> should have been denied boarding since the airline would have a transportation liability (airline pays for the flight back, since they didn't check the documents properly, and would be fined additionally).
b) Assuming the child is a US citizen, entering on a foreign passport (fine)
c) Entering on a non-immigrant visa (since intent to stay but not even ESTA). The OP was lucky the immigration officer let this fly. Depending on his/her mood, he could at least made a night or two in immigration detention center uncomfortable for you  until you see a judge. Although unlikely a judge would deport a child with its unclear status and the current situation, spending a night or two in one of these centers with an infant would be very very uncomfortable. 
 
@makhan27 you have no idea how lucky this went for you considering all above. 

Any US citizen can hold a US passport, that was not my question: Does she fulfill the physical presence requiremts inable to transfer her US citizenship to the child? If not, you might be facing a serious issue. 

I agree, they were lucky in so many ways:
a) No valid ESTA -> should have been denied boarding since the airline would have a transportation liability (airline pays for the flight back, since they didn't check the documents properly, and would be fined additionally).
b) Assuming the child is a US citizen, entering on a foreign passport (fine)
c) Entering on a non-immigrant visa (since intent to stay but not even ESTA). The OP was lucky the immigration officer let this fly. Depending on his/her mood, he could at least made a night or two in immigration detention center uncomfortable for you  until you see a judge. Although unlikely a judge would deport a child with its unclear status and the current situation, spending a night or two in one of these centers with an infant would be very very uncomfortable. 
 
@makhan27 you have no idea how lucky this went for you considering all above. 

While I agree with the CRBA being the best choice, if the OP fulfils the physical presence requirements, he can just do a DS-11 passport application at a DOS passport agency. 
 
Like I mentioned a couple times above, you can go to a DOS passport agency and just apply with the DS-11. No I-551, no 181 or CRBA needed. The passport will be issued directly by DOS and you have consular officers working at the agency familiar with immigration law. If you don't know what a DOS passport agency is, see here:
https://travel.state.gov/content/travel/en/passports/get-fast/passport-agencies.html
 
You have time until the child is 18 y/o to do a CRBA. If you are traveling back in December that might be an option and you would get the FS-240 that is important. The process takes less than an hour and the CRBA and (emergency) passport can be issued within an hour if needed. 
 
It has some risks though: The consulate requires you the proof of physical presence of the 5 years. If you should not be able to provide this during your vacation, no CRBA and passport will be issued. So in theory your child was illegally in the US on ESTA with an overstay and you might spend longer in Europe than you wanted to and won't be able to travel back. 
 
To sum it up: Better option for whole life will be CRBA (with FS-240 document), safer will be getting the passport now. 

Situation B:  My child was born in wedlock, out of a U.S. citizen mother and a non-U.S. citizen parent: The U.S. citizen parent must present evidence she or he has been physically present in the United States for a minimum of five years total, two at least must have been after age 14.
https://de.usembassy.gov/u-s-citizen-services/citizenship-services/crba-checklist/proof-of-physical-presence/

Any US citizen can hold a US passport, that was not my question: Does she fulfill the physical presence requiremts inable to transfer her US citizenship to the child? If not, you might be facing a serious issue. 

Does your wife fulfill the physical presence requiremts? 

How was the child able to enter the US? 

Not necessary, they are giving you false information. Where did you apply? A passport office, or a DOS passport agency? Remember the person working at a post office, won't be too familiar with immigration law. Look further below:
 

How was the child able to enter the US? Under which visa and category?

Simply because it is easier and costs less: A CRBA is necessary to proove to USCIS that there is no claim to USC. After that, a BTL can be issued. BTL's are processed on the spot and can be issued within a couple hours. I-130 will take months, if not over a year and cost way over a $1000 for fees and medical.
The fathers US IV is already issued. If they go through the I-130 process now (which is totally unecessary) the fathers visa will expire before the child will even have an interview.