Lil bear

Possibly yes.  Not guaranteed 
 
Yes. 

Adjustment of status will always be of the table. If the undocumented person 
marries  a USC, this  will then allow for the petitioning of the non US Spouse through the Consular process. The interview will be completed in their home country. Departing the US for this interview will trigger a ban of up to 10 years.  The immigrant visa will be denied and the applicant can then apply for a I601 waiver. All this would  happen with they remain out of the US .. and will take 2-3 years 

Reread the replies.   No.  It will not be accepted. Even if it was placed on the file,  a new one will be requested after the application is denied. 
 
They can’t ask for a waiver until the visa is denied for an incomplete medical.   The waiver will add an extra year to the overall process.

Adjustment of status will always be of the table. If the undocumented person 
marries  a USC, this  will then allow for the petitioning of the non US Spouse through the Consular process. The interview will be completed in their home country. Departing the US for this interview will trigger a ban of up to 10 years.  The immigrant visa will be denied and the applicant can then apply for a I601 waiver. All this would  happen with they remain out of the US .. and will take 2-3 years 

https://www.uscis.gov/sites/default/files/document/forms/i-864instr.pdf

If you ever has an A number .. which of course you did.. put it in. They already know it .. and they know you have an A number. 

She should relocate to the US. Then you submit evidence of her arrival and domicile , get your visa and go.

Having entered legally into the US the last time the person enters is a requirement for Adjustment of status if eligible as an immediate relative.   No legal last entry .. no AOS possible. 

Having entered legally into the US the last time the person enters is a requirement for Adjustment of status if eligible as an immediate relative.   No legal last entry .. no AOS possible. 

How did they enter the US ? Visitor visa and overstayed ?
 

Correct terminology
 
K1 visa is a single use visa granting entry upon presentation
Entry using K1 results in a period of “authorised stay” of up to 90 days  
After the 90 days, the K1 user becomes “out of lawful stay” status, and is subject to deportation, unless a further period of “authorized stay” is granted by application such as Adjustment of Status. 
 
Accurate understanding and use  of terms is really critical in these processes   Many many delays denials  and refusals occur because of inadequate understanding  .. it gets really disheartening and very expensive when this happens 

You can ask but it’s in the hands of the Consulate and may not get you as much time as you need… you may still find yourself running out of time to enter with an issued visa or being very delayed in interviewing  . I re read your post and see that the I 130 have already been filled for your children 
 
You can still  follow the rest of the info in option 2 .. 

Your options
 
1. Your petitioner should  file I 130 immediately for each of the children. Slow down the submission of your NVC  documentation until their petitions are approved and sent to NVC. Request the NVC to link the cases and continue the processing together. Immigrate together 
 
2. your petitioner should file I130 immediately for each child.  Continue your processing through NVC and consulate interview without delaying it. Arrange for your children to stay with relatives in your home country for about 1 -2 months. Once you have your visa, enter the US.. Your stamped passport will be your  evidence of your LPR status. Immediately get your SSN, bank account etc as evidence of your residency and intent to domicile in the US.  Immediately file for a Reentry permit I131 for yourself. On the form, indicate that you want to pick up the issue permit at the US consulate in your home country  Wait until you have received notification of the receipt of your application and either have had on file biometrics applied or attended the required biometrics appointment. You can then leave the US and return to care for the children while their immigration process takes place.  It may take 6-12 Months for the permit to  be issued. You have 2 years from that date of issue to return to the US without being assessed as having abandoned your LPR status. Return yo the US using your reentry permit, accompanied by your children using their immigrant visas 

Possibly yes.  Not guaranteed 
 
Yes. 

Possibly yes.  Not guaranteed 
 
Yes. 

IMO, the I131 will be rejected as the requirement is that the applicant must be in the US when filing for the travel permit .. not that the I131 must be “filed/posted”  from within the US. I recall a thread recently where an OP was in the US, mailed the I131 application, and left the country the next day .. before the USCIS received the application .. and were denied because they were deemed to have not been in the US when it was filed. 

I used the stamped visa. It is valid for 12 months from entry. No need to wait for the GC and no need to risk getting it mailed to you 

I used the stamped visa. It is valid for 12 months from entry. No need to wait for the GC and no need to risk getting it mailed to you 

I used the stamped visa. It is valid for 12 months from entry. No need to wait for the GC and no need to risk getting it mailed to you 

I used the stamped visa. It is valid for 12 months from entry. No need to wait for the GC and no need to risk getting it mailed to you 

They don’t care who actually puts it in the mail. Just make sure it is what  they are asking for, has the correct cover letter attached, and is sent too the correct address. Use a signature required, proof of delivery Service and keep photocopies of everything sent 

It would put you back to the start. Visa issued but not used is voided after the expiry date 
 
refile, pay$$$$ again and  go through the WHOLE process again 
 
Follow CrazyCat’s  advise 

The OP had separate AP and EAD cards rather than a combined one. Also, some people have applied for a reentry permit during their time as LPR, so that is also needed to surrendered at naturalization 

The red section is to be read in conjunction with the upper section. Crazy Cat is correct .. if the document is in Spanish it does not need an english translation 

Yes we had that. No employment history. It is based on answers to earlier questions .. though i cant tell you which ones !!