I am not sure if it will be like this for everyone, but let me share with you what it was like to "co-parent" with a spouse who cheated on and divorced me. If you remember, in many ways our stories have similarities in that I married for life, I wasn't blind but I never even thought of other men, he was "flirty" and then one day I got the absolute proof of infidelity. Like you at first I somewhat freaked out, and then I started to get my own head together, got myself and my life in order, started counseling and reading about marriage and support groups, and began to see the patterns he refused to acknowledge. Eventually, rather than address his own issues, he chose instead to leave. At the time I was mid-30s and my kids were on the younger end of elementary school.
I know for a fact that when my ex left he thought I would be around his little finger to "control from a distance" while he did what he wanted with who he wanted. I know he also expected I'd continue to meet some of wifely needs such as "free babysitting" and that I would financially care for the kids and not expect him be a responsible parent. I think he suspected he would get the kids and I would care for them ... for him.
I know his version was basically "continue as we are except that I move out, don't have to put up with you, and I can get laid by others" LOL
Imagine his surprise when I *DIDN'T* ask for him anymore, didn't call him constantly, and didn't want him involved in my life...at all! He did, at first, completely take off for a few months but thereafter he would subtly try to get back in or try to force me into a corner, but I wasn't interested anymore. We ended up selling our marital home and we moved 3 miles apart. I had the kids and *DID* put child support into the divorce, and it angered him that I asked for it (naturally it was me being selfish, not society expecting him to care for the children he decided to create). And he was into his affairs and running around enough that we had 50/50 in the divorce but in real life we worked it out so the kids lived with me--just 3 miles from him--and they could come by him any time he arranged it or they asked to see him. I think they got together about once a month. He didn't even have a room for them to sleep over for several months!
At first he did try the "You need to do this for the kids..."
but when I didn't fall for that, he did eventually stop. When I didn't return his phone calls or emails, he did eventually stop them too. When he'd ask where I was, I never told him my plans or who I was with or why--just "I have something scheduled. That won't work for me" or "I have an appointment so can you watch the kids or shall I make an alternate arrangement?" and that's it. When I saw him, I didn't go linger in his house or try to "talk" to him--just dropped the kids off in the driveway and left. When he dropped them off I didn't invite him in. I was civil and courteous as I would be if he were the pizza delivery man, but that's it.
Now he lives on one coast, I live on the other coast, the kids are both grown and he periodically calls them on the phone. They both live near me. And for graduations and things, they include him, he "plans" them from thousands of miles away, and everyone pretty much agrees to just put up with him for the couple of days he's around. He has a new life; I have a new life; and neither one of us really have any ill will--we just don't speak.