I would lean toward keeping H in the house and accepting his last ditch effort to be honest as he showed you things he did not ever plan to and did not have to. Don't get me wrong as there is a lot of work ahead. But, you have to choose forgiveness if not for him... for you. Please set boundaries on what is acceptable behavior but you are going to have to get to a point where you just don't bring it up again. Could you? Could you forgive and not bring it up again?
I think in reading through your posts I sense that keeping H in the house gives you the feeling of control.
If he moves to an apartment you lose control. The ability to know if he is in touch with AP. Control comes from fear --- so what are you afraid of? You've listed it. Your fear is if he moves out he will be contact with AP... but wouldn't it mean more to you if he moved out and he made a choice to not contact AP and instead chose to do some work with the idea of winning you back? I think that is a much bigger powerful statement than he stays in the house and commits to you under duress.
There is no easy answer here.
You have to chose what is best for you. Try taking your kids out of the equation for a bit. Sure a 2 parent household is best but a household where mom is miserable or dad is miserable is still not a healthy environment to grow up in.