I'm sorry that things are a bit crazy at home. You need to decide what boundaries that you want to make to help you keep grounded. If it means that you need to change up him coming in the house while waiting for your son, then do it. Your son is old enough to be ready by the time he comes and then walk himself outside. Your h doesn't need to come in each and every time he picks up son. Your home is now your safe space. Would you be able to do this at his place? I seriously doubt it.
As for him doing things around the home, you will need to decide what you and your son can do w/o his help. If your h offers to assist and you feel comfortable w/him doing things, then okay and be sure to say thank you.
What I see is a man who has made a choice not to live at home w/his family. He may be confused, but he knows exactly what he is doing and that it is his choice not to work on his relationship w/you at the moment. He's keeping his options open in case the ow situation doesn't work out and you are now considered Plan B.
He's not going to miss his home, you or his son if he continues to use the revolving door of his place and yours. I do think that he feels guilty for what he's doing and that could be why he's been doing some chores, but I would certainly think about setting some boundaries. How can he miss what he sees all of the time?
As for having discussions w/him about the relationship...no more. He's not going to work on the marriage as long as the ow is in the picture. His focus is on her and their time together. Start, in a nice way, curtailing some of those 20-30 minute visits in the home. You might even want to say, "h, how about calling ahead and letting son know what time you are planning to come by and he'll be ready to go when you get here and will meet you at the door so that he doesn't take up a lot of your time".
It's okay to be friendly and supportive, but that doesn't mean allowing them to have their cake and eat it too. Time to set some boundaries for YOU!
Sit quietly, the answers will reveal themselves when you least expect them to. The past is gone, the present is a gift and you need to focus on today, allow the future to reveal itself when it is ready.