I went back and read some of your posts in other threads and wanted to add to what I'd written before.
I'm actually quite happy to read that you've been in counseling and are actively working on becoming a better person. I think the world would be such a better place if EVERYONE tried to do better today than they did yesterday and I appreciate that you're doing it.
So if you want to hear my take on all this, read on. I'm no expert, so take what you like and leave the rest.
At this point in your situation, your wife is in bad shape. Is she going to counseling? If so, I hope it's with someone who understands infidelity and trauma. Helping her to heal will help the whole family.
If you've agreed on the financial terms, I'd suggest you give in on the custody part but have the right of first refusal when the kids need overnight care. Then just let everything happen but be kind, be generous, be remorseful, admit to the kids that it's your fault and that you're very sorry and that you promise to never let them down again. And then, just do it and let time perform it's magic. Right now, she can't trust you. Not because you're not trustworthy but because she isn't able to trust anyone. I guess what I'm saying is, Be part of her solution, not part of her problem.
And whatever you do, don't even mention her to your kids unless it's in the most generic way - did Mom go to the store? That sort of thing. They'll form their opinions and judgement about both of you based on their own experiences. Just be kind, offer to help when you can see it's needed, proactively take out the trash, etc. without bringing attention to the fact that you're doing it. IOW, do it without any expectations of praise or thanks. Humble pie to the nth degree. That is designed to help her feel safe and to help her not get her hackles up at the sight of you, which is probably the case right now but that doesn't have to be permanent.
I played games pretty much non-stop during the last year of my marriage and for 2-3 years after the divorce, although mine were the match the dot type, not what real gamers play. My mind was mush, my brain was paralyzed and I could do little else. So perhaps be a little more understanding of her foibles right now and refrain from criticizing her, even in your mind.
And, last but not least, spend some time reading the Just Found Out forum. Pick a couple of the really long threads. It'll help you realize the level of pain caused by infidelity. My pain is no longer making my stomach flip all day everyday but after 4 yrs., I still don't care if I wake up tomorrow morning. This pain is real and it's really, really hard to get past. Granted, I was much older and had been married for 45 yrs. so it totally knocked my socks off. But you're younger and there's hope for you and your family. And I hope you realize that it's that hope that has caused me to sit here and type all this.
BW, was 67; now 74; M 45 yrs., T 49 yrs.DDay#1, 1982; DDay#2, May, 2017. D July, 2017