*sighs* --- in relation to the first comment that was posted --- the world is not made in the image of Western countries. We are dealing with a country where custody orders would be exceedingly rare and most people tend to practice according to custom, rather than modernized law, because almost nobody is motivated to actually go through any type of court process, and the cultural custom is that for married parents, fathers are assumed to have 100% custody rights, and for unmarried parents, mothers keep their children, and there basically are next to zero child support cases.
Put simply, and without going into unnecessary personal details, there's good reason to know the biological father just isn't going to be in the picture.
So I would appreciate if comments only focused on the immigration issue going forward, with people who have direct experience with the immigration process in situations like these.
Redro --- thank you for your response. To answer your last question first, this is a country where the case will be handled by an embassy that almost never has K-1 cases, so there are almost no cases to go on, which I know may not be extremely helpful, but I always approach everything with honesty, and I intended to go to the interview personally to do whatever I can, unless the embassy just says they vastly prefer against that.
It is so rare for a K-1 visa to occur in this country that the official stats show that there is only one in process, which I assume is mine, so for privacy concerns I would rather PM you, if that's okay. Let me know if that's okay with you.
On your first question, the birth certificate isn't fully processed yet. I'm not totally sure what she will do about indicating the father but she's very likely to give the child my last name. The problem is that the passport application for this country asks about parentage, and we aren't super clear how that works for a "father unknown" situation --- we have good reason to think that putting that would be inadequate. But, on the U.S. side, we obviously can't claim the child is a USC, so we aren't sure what to do.