Dude, I told you she's not over you...

I would totally call her on the adultery and tell her she's already been cheating on you and you caught her TWICE. Tell her she is no position to be trusted right now.

I would talk to your lawyer about her showing up whenever she wants to.. I would tell her unless she's prepared to surrender a key to HER new place that she can only come over if she makes an appointment.

You sticking to her not being allowed over will send a very clear message to her that you are not clinging...

She thinks she can use you like a doormat. You could trap her and tell her OK, so if I catch you with any other man in the next three months even flirting then you will turn the house over to me completely?

Signed sealed and delivered full ownership?

Ha ha... she's walking right into a trap that route.. but you would have to get her to sign something she would never admit she made that agreement otherwise...