I travel from the US to international sites once every couple months. I have no idea at what point, if any, you are checked against the Do Not Fly list, except perhaps when you buy the ticket. I have had to provide my number before the ticket was issued in two cases I recall, so probably then, but once I was in the airline's system I didn't have to do it again. When leaving the US, I show my passport when I check in with the airline. They use it as my photo ID, and the only thing they might do otherwise is confirm that I have a valid visa for the country I'm going to, or that I have a full blank page for the stamps. I show it again at security, and I might have to show it again at the gate, again to confirm visa and blank page, but all they do at any of these is look at it, or put it under UV to see the hologram. Same for boarding; all I have to do is show it, which is not always. US citizens leaving the US do not have to complete departure cards or go through customs when they leave. The passport is not scanned at any time before leaving. If I understand your scenario correctly, it would be easy-peasey for someone with a fake US passport to leave the US. I suppose someone could book the ticket with a non-flagged passport number then fly on another passport. Hopefully there are more safeguards than I know about.
Entering the destination country, however, is another matter. Even in the poorest African countries I visit for my job, the entry is pretty sophisticated. First, you fill out an entry card, then wait in line. At the kiosk, your passport is either swiped along the long edge or the photo page is placed on a glass scanner. Here is where a fake passport would be detected. If it's real, the normal procedure is to take your picture with a webcam camera, and scan your fingerprints on a scanner with a screen about the size of a playing card. Sometimes they want all four fingers, sometimes just the thumb. Such places are noisy, and especially in non-English countries the agent just holds up a thumb to indicate to scan your thumb, or all four fingers for that. Sometimes they ask how many days you are in country, why, and where you are staying. Once at the window, the process takes less than a minute. I haven't had this happen or seen it happen, but if the passport isn't recognized by the system I suspect the person would be detained until either the embassy cleared it up, or the person is refused entry and placed on a plane back out of the country. Of course, for story purposes these officials can be believably bribed.
ETA: if your character is sent back to the US after his passport is denied, he's in deep when he arrives and tries to re-enter the US. Chances are he will have problems gettkng into any country at all.
Probably more info than you wanted, but that's been my experiences.
