I'm so sorry you are going through this. It truly is awful.
It will be so much better once he leaves. H was only at home for a few weeks after BD, and it was horrible. I was sad when he left, but that quickly turned to relief. The space will do wonders for your mental health. And honestly, if he is with OW, he already left in his mind. He's a jerk for trying to make you do the 'pick me' dance.
Good job on keeping quiet about the OW! It is hard having to keep your intel a secret, but it is smarter for the long run. I'm in the same situation, it is furiating to see him looking all smug and knowing he thinks I'm stupid. Do you have some close friends and family members that you can confide in?
As far as legalities of him moving out -
If the car is in your name, it should be yours, even in a community property state. (both of our cars are in H's name) Especially if you aren't married, it is yours. You could call and report it stolen if he takes it without your permission. If you need proof of adultery for legal reasons, you are allowed to put a GPS tracker on the car if it is yours (confirm with a L).
Talk to a L about child custody and be careful about setting precedent. Keep the kids at home with you until you get legal counsel. I let H take the kids overnight, and now my L says I probably can't take away overnights because I already let him do it (grrrr). I regret letting H bully me into a schedule before I talked to a L. Make a formal schedule and stick to it. Pick a schedule you like, he's the one that moved out. Research standard schedules where you live. Let him see what a divorce will look like. If he doesn't want to be with you, he needs to experience not being with the kids 50%+ of the time. That's divorce life. Don't sugar coat it for him. He is firing you as his spouse - it is no longer your concern if he is happy or not. Do what is best for you and the kids.
Whose name(s) is your house under? Talk to your L about not letting him back in the house after he moves out.
Is there legal separation where you are? Taking legal action seems like you are giving up on the marriage, but you have to protect yourself. My H moved his paychecks to a separate account and drained the joint account after declaring that he wasn't going to financially screw us over. Trust nothing that they say. They are selfish liars and you need to take care of yourself and your kids.
The situation is a nightmare, every day you get through it is a success! Be proud of yourself! You are awesome!
