You are right from the US he would need your permission, not sure about where you are going or for entry into the states. And it has nothing to do with custody rather than rules about children traveling out of the country without both parents. Also not sure how it works if he uses the non-US passport. INS/Customs/TSA etc won't have record of custody arrangements.
So he expects a casual boyfriend to pay your bills. GRRRR. Again, I think he expects you to meet someone so he doesn't have any obligation.
I asked about Europe because you may need something to use as leverage for negotiations. That is all. I know you are going but right now he seems to hold much (all of) the power because you have done everything he has asked. If you push back a little bit, you might find that he is actually more cooperative because he wants to get to a certain end game. (you in Europe, for example)
I know you plan to consult an attorney before you sign. Try your hardest to think of everything you need to include.
Remember in all likelihood the OW is advising/pushing him. And as others have written you cannot underestimate an OW who is trying to win a "prize."
