[Solved] Dealing with a lying, cheating, mother to my kids

So two weeks have passed since the mother of my children left me. She told me that i was mentally abusive, she hated every time i touched her, and that she felt like i never care about her. She also had this friend, a guy from her work, that she had been talking to a lot for the previous few months. I asked her if she cheated on me with him, and if she was leaving me for him. She told me no, they were just friends, werent interested in having [censored] the whole shabang. I knew it was a lie. She dumped me on a Tuesday, the previous Saturday she slept with him. Her hip hurt on Sunday from it, i tried to massage it, and when she wouldnt let me touch her i knew. Long story short. Shes since admitted to being a sadistic liar, that she doenst know much about her outside of all her lies, that she does it because thats all she knows how to do. She now says that she wants to be with him for a really long time. We both made mistakes, neither of us were all that great to the other. But this is the 3rd time in two years that shes cheated on me, and then left me. Yes i kept taking her back, no i never forgave her and held it against her. This is the first guy shes ever wanted me to meet, or try to be friends with. Shes told me hundreds of times over the past 6 years that she hates commitment, hates emotions, and that she only has relationships so that she doesnt feel lonely. Just over 5 years ago, we broke up for the first time right after my oldest son turned 1. I was upset, and angry, and i threatened to take him from her because she has mental issues and because the house she lived in was completely trash and horrible. She ended up running and hiding my son from me for 2 years. We got back together when he was 3, and a year later we had a second son. We were trying to be together because of the kids, and because we actually did love eachother.
The reason I'm on this forum is because of what she expects of me. She moved out a year ago, and we've been working on things living apart but being exclusive to work on our friendship and to be parents. Now that she has this new guy, who has been staying at the house with her and of course my children, probably since the sunday or monday before she ended things with me, she expects me to be okay with what shes done to me, with absolutely nothing from her aiding in closure. She doesn't want to have the kids bouncing back and forth between homes, which i can agree with. But what she expects, is for us to still be a family every weekend, and the kids live with her. The issue im having is that i cant stand being around her for the hurt shes causing me. I've told her that the only way i can make that work, is if she puts her thing with this guy on hold until i'm more comfortable. She says she wont because she wants to be with him for a very long time. Shes offered to limit the time he spends at the house, and for me to come over whenever i want, of course upon her approval. Shes even offered for us to go to counseling to become better friends and parents together. She wasnt willing to go to counseling to make our relationship work, so why is she now? If she really liked this guy, why is she willing to do all this to him and to me, just so that the kids will stay living with her. I feel like she is just using him to hurt me. I feel like shes doing all of this so that i will screw up so badly that i will never be allowed to see my kids again, or that i will just give up from the pure suffering i am going through because of this. I feel that since she doenst want to be seen as the mother who has ran and hidden the children twice, that shes trying to do everything she can to just make me give up. There is so much more to this story, but im not thinking very well at the moment to be able to put everything to words...