None of that would matter in NYS family court, your state must be different. Cheating, lying to your spouse, being a 'big' person and not talking about the mom or the dad, wouldn't play any part of custody, and neither would having an affair, unless you were having it in front of the kids. You can live with someone, and as long as he treats the kids well, and isn't a sex offender, that won't play into custody decisions either. Treatment and support of the children, come first and foremost, along with a 'steady' environment. She (well, TLC) has Jon on tape saying he doesn't like staying at home with the kids..back when he was working and recently. He was out in a bar at a time when we assume he was supposed to be taking care of the children (although that won't mean diddly if he had a sitter). I can see no reason why he would get them in anything other than a shared arrangement.
But Jon doesn't strike me as being very savy, so if he knew something, I doubt he would keep it quiet. I still can't get past that so many are willing to close their eyes on what Jon has been up to and either say that Kate had an affair, so it's only fair (or she deserved what she got), or that it just seems to be OK that Jon cheated on his vows, but it wouldn't be OK for Kate

.