JimmyHou

I have to agree with the previous posters, this is really over our heads and you need a good lawyer. I wonder what could have happened to make them schedule her ceremony and let her go along with it and then change their minds at the last moment.
Your point about her being a minor and not responsible, is probably right; she's not responsible for any wrong-doing, however, that does not guarantee eligibility for citizenship. Let me give you a unrelated example. If the father and daughter were both permanent residents and the father got a job abroad when the daughter was 12 years old and they both moved to Europe for 6 years without returning to the US, the daughter would have a very difficult time coming back at age 18 because she would have been out of the country for too long. Even though she was a minor, unaware of the rules, and subject to her father's decisions, the same eligibility rules would apply to her.
Having said that, your case is much more complicated and a good lawyer may easily find a work-around. I really suggest you consult one. Good luck to you, you're in an unfortunate situation through no fault of your own or your wife's.

I know it's not what you're hoping to hear, but Silkafi88 and Shub are right.
You have to wait before you apply and even then, you'll be questioned about why you were gone for so long.
You should not apply now; your chances of approval are very small.

Actually, there are a couple of cases I've read about, where it was needed. Both involved a discrepancy between the name on the green card and the name on the passport. If the two names are not exactly identical, the interview officer will ask to see either the birth certificate or a legal name change document (or marriage certificate in case of a last name change only) to determine the legal name.
That's the only situation I've come across where the birth certificate was needed.

Here's your answer - http://www.uscis.gov/us-citizenship/citizenship-through-naturalization/continuous-residence-and-physical-presence-requirements-naturalization
Read it carefully. There will be no more answers other than ones given above.

You have to have been a resident in the state in which you are applying for at least 90 days at the time you apply. If you are applying in Florida, you have about two months to go before you can send in your application. You should be prepared to show some sort of proof of how long you've been in Florida if asked.

If you're field office is Baltimore, then you're right; they do offer same-day ceremonies:
"The Baltimore Field Office conducts same-day naturalization ceremonies, but there is no guarantee that your case will be eligible for such a ceremony on the day of your interview."
http://www.uscis.gov/about-us/find-uscis-office/field-offices/maryland-baltimore-field-office#Naturalization Ceremonies
Sorry, no information on guests.

All of the answers above are really good ones. Ease of travel is an obvious reason for anyone without a passport that gets you into most of the world without a visa. Permanence and stability is another one; for hose of us who no longer have ties to your former country, then why not take the plunge and become an American? I plan on maintaining my current citizenship, but others may not care.
Another reason is freedom; I know, this sounds like a cliche, but everyone on VJ comes from countries with varying degrees of civil liberties and oppression. Some people on here got their green cards through asylum and have gone through things that make them want to forget their home countries. Others love their home countries but know they can't go back. People who would be discriminated against on the basis of sexual identity or gender or race or religion may feel safer with a US passport, even if they travel back home. In fact, even without discrimination, many countries treat Western citizens better than their own people as soon as they land at the airport. I'm not saying that the US is discrimination-free or that there aren't countries that are further along in this area than the US, but the US does offer a good balance of freedom, economic opportunity, and a reasonable immigration process.
Finally, I think some people just fall in love with this country; I know I'm one of them. The political system is broken, there's a lot of anger, there's corruption and gun violence and crumbling infrastructure, but it's still a country that took US in when it didn't have to. I was allowed to study here, to get a job, to buy a house, to travel freely, to settle permanently, and to apply for citizenship hopefully soon! The US hasn't handed us anything on a silver platter; everything's been a struggle, but the country has offered us an opportunity to work our butts off and see how well we could do. Citizenship is just own more opportunity to work towards. The citizenship oath is oddly worded and anachronistic, but a lot of people on here take it seriously and a lot of us wouldn't think twice about fighting for our adopted country. I honestly think that if I didn't feel this way, I would just keep my green card. It's an inconvenience having to get a visa to go to Europe, but it takes 3 weeks to get a Schengen visa and the UK is pretty generous with multiple-year visas, so I can handle the inconvenience.
So I guess for me citizenship is a way of saying I've enjoyed being a guest in your club, and I've decided to become a member.

I assume that this has come up on Visa Journey before, but I think that some of you who are about to apply or who are in the middle of the application process will find this helpful.
I'm sure that almost everyone on Visa Journey has read through the USCIS guide to naturalization (see link below) because it does a decent job of describing the application process.
Document # 1: USCIS M-476 "A Guide to Naturalization" http://www.uscis.gov/us-citizenship/citizenship-through-naturalization/guide-naturalization
However, I'm not sure how many people have looked through the USCIS Policy Manual. Volume 12 of this manual (see link below) gives guidelines to USCIS Officers (not to applicants) on what to ask an applicant for naturalization and who to approve. Many of the questions I've seen posted on Visa Journey can be answered by reading through this (rather long) document.
Document # 2: USCIS Policy Manual Volume 12 "Citizenship & Naturalization" http://www.uscis.gov/policymanual/HTML/PolicyManual-Volume12.html
Good luck everyone!

Songer,
I'm not at the interview stage yet, but I came across some posts that you may find helpful.
Here's a case where rescheduling the interview only caused a 60 day delay:
http://forums.immigration.com/showthread.php?323930-N400-interview-rescheduling-any-experiences
Here's a case where the applicant asked for a rescheduled interview, but didn't hear anything for a weeks afterwards (unfortunately, we don't know how this one ended):
http://www.trackitt.com/usa-discussion-forums/n400/1147187227/reschedule-naturalization-interview
There are three applicants on this thread who rescheduled their interviews (with delays of only 1-3 months):
http://www.visajourney.com/forums/topic/210024-postponing-naturalization-n400-interview-date/
Here's a case where USCIS rescheduled the applicant's interview for a later time on the same day:
http://forums.immigration.com/showthread.php?242615-Rescheduling-citizenship-interview-experience-at-the-Minneapolis-St-Paul-district-of
Here's a case where an applicant (not the original poster) sent in a rescheduling request that was never processed and had to explain why he/she missed the interview:
http://www.visajourney.com/forums/topic/183080-rescheduled-your-n400-interview/