CQ & CS

The inconsistent answers to the questions about her kids is not the reason you are in this mess. They conducted a Stokes interview, which means they had serious doubts before those questions were asked. They didn't view any of the evidence you brought, so there is NO WAY that this suspicion arose at the interview. They had strong suspicions before the interview ever began.
I strongly suspect that her first husband is the reason for this. You said he got his citizenship through her. He also has custody of her kids. Fathers don't usually get sole custody of the kids in the US unless the mother is found unfit, or she simply has no emotional connection with the kids. If there are any other unusual circumstances with the first marriage, like if they divorced before he got his 10 year green card or if there was a VAWA claim involved, then that would only compound their suspicion.
They focused their questions on the kids because they want to know how a woman could live with her new husband for more than a year without having regular contact with her children, and without her new husband also having regular contact with them. This situation makes it look like she had no emotional investment in her previous marriage, and therefore no emotional attachment to the children produced from that marriage. In other words, they suspect that first marriage was a sham to help her first husband get a green card, and now she's interviewing to help her second husband get a green card. 2 + 2 = ???
Your attorney has failed you miserably. When she learned about the circumstances of the first marriage, and the fact that immigration was also involved in that marriage, she should have asked a million questions and collected as much information as possible about that marriage, the immigration benefits involved, and the circumstances surrounding the divorce. She should have also known that a second marriage to an immigrant would be scrutinized by USCIS, and she should have prepared both of you for the very real possibility of a Stokes interview.
You need a new attorney. This time, you need to grill the attorney about their experience and track record with similar cases before you write a check. Do a little basic research into the law, as it relates to your case, and you'll be better equipped to ask questions that will help you determine if your attorney knows what they are doing. A good attorney will be expensive, but it will mean the difference between success or deportation.

Not precisely. Getting married within 90 days is the only requirement in order to be eligible to adjust status based on a K1 visa. It doesn't mean you are "legal" if the I-94 expires. "Out of status" means you have no lawful status as defined in the INA - you're not a non-immigrant, you're not a permanent resident, you're not a citizen or a national of the US. You're an alien whose permission to be in the United States has expired. What separates you from someone who overstayed their visitor's visa is that you would be eligible to adjust status, even though your I-94 had expired.
As Haarp425 stated, you begin accumulating unlawful presence when the I-94 expires. You stop accumulating unlawful presence when the AOS petition is accepted, but that doesn't restore you to lawful status. What it does do is grant you permission to remain in the US until a decision is made on the AOS petition. In other words, you're "out of status" but not "unlawfully present".

Do you mean US passport? You can just take the picture in front of a white backgroud and send it to Walgreen to print. This is the cheapest way. Of course you have to pay attention to the pictures requirements.

Sent my RFE yesterday. Just so anyone who gets RFE knows, my RFE was sent to a PO Box in Missouri and Fedex/UPS won't deliver to a PO Box; only USPS does and it takes two days even for Express Post.
I put that in my signature for heads up.