Gonna go ahead and say the unpopular thing here: it's entirely possible for a person to still be friends with an ex. Not all breakups are bad, sometimes you figure out you're not right as a romantic partnership but it's still possible to be good friends. Also, you've only been dating for 6 months. She's probably known this guy a lot longer than you, and if you push too hard you may find she cares more about her friendship with him than she does about the relationship with you. So be careful not to come across as too controlling. I'd avoid asking her to cut off all contact with this person, etc (assuming you're trying to salvage the relationship, which honestly might not be worth your time.)
That said, letting him sleep over at her place (assuming it wasn't offered to other people as well, e.g. after a party when it's late, people might be drunk, etc) and live rent-free in her property are a big step too far, in my opinion. And the threats and sour cream thing (?!) are definitely off base on his end, and she should have objected to him about that. But getting upset that they text or hang out says more about you than it does about her.