Right, so correct me if im wrong. You did live together but he moved out due to your mood swings, you have kids together too but he hasn't seen them....
I would seriously have a word with this guy, this isn't a relationship, you clearly want different things to him. From what you have said, it doesn't surprise me if he's still has some involvement with the woman he cheated with.
Yes, its easier to say that your partner dislikes her and calls her names but that doesn't mean anything. Couples stand in church when getting married and say their vows and it means sod all to some of them as they still cheat.
Why isn't he unfriending this woman from social media?
For you: Whats the point in carrying on if this isn't the relationship you want. Think you need to have a serious word with him and ask him what he wants out of this relationship. Do you have a good relationship with your partners mum. Is it worth talking to her?