Personally I would have instantly just broken up instead, but he was on the hook for the apartment lease so I guess he preferred this way. Either case she got what she asked for and OP is in no way to blame.
Yep. Once a woman suggests that, a true relationship is done. You can't have significant trust in someone that sleeps around and it makes long-term relationships or marriage impossible with them. You can date/have fun but that's all it will ever be.