I've read the whole thread, been following since the beginning. I also saw your other thread about him needing space, OP.
I'm really sorry to say that as an objective observer, reading all the facts you've posted here, the evidence really does point very strongly towards his having cheated on you.
It was probably a one off. He probably took the opportunity while he was in the states thinking he could get away with it and you'd be none the wiser. But he didn't cover his tracks and now you have a seed of doubt in your mind that will only grow over time.
It might well be that he feels terribly guilty and wishes he hadn't done it (to wit: the erection problems). And if that's the case then maybe, if you want to, you can work past it.
But I don't think he's being honest with you right now. And I don't think you should move in with him. I think it would give him the message that he can get away with it. That you're willing to turn a blind eye now and so therefore will be likely to turn a blind eye to similarly ambiguous evidence in the future.
At least if you put your foot down on moving in together now and insist on taking 6 months to be absolutely sure you want to continue this relationship then it will be a signal to him that you won't take any shit.
Sorry you're having to deal with this OP. When he's the one that's fucked up the trust between you.