Just catching up. And many congratulations on the young Sapling! You sound very well, all things considering. 
Maybe it's because I was reading your update on the way home from a huge difficult meeting I've been preparing for all weekend, but your update makes me outraged on your behalf!
And you are still so considerate. Don't know how you do it. If the OW is a MNer, then ODFOD to the nth degree.
But seriously, I think this the moment to stop being so caring & reasonable to your WH. He's obviously still in touch, even if at second-hand via his gaming friends; he obviously still has a connection to that world, and is in some way yearning for it.
I think he needs to choose. Unequivocally.
And he needs to know what he is putting in the balance of that choice. You've managed well without him (let's leave aside the huge emotional loss you will/are inevitably feeling)
I'm a list-maker, so here's my list.
- Don't do the Pick Me dance. You don't need to.
- He needs to stay away, to realise what he has put in jeopardy. Let him be with the children in a typical EOW and overnight during the week. And not in YOUR house.
- Or - let's be modern here - let him have full responsibility (child care & all) of his two boys (not the new born) 50% of the time. Let him sort them out in his own home.
- Is he still gaming? He is the father of three children and he still spends time in a mindless pursuit like that? (Do you get equal time to do your hobbies? OR just sit and do nothing? Really?) Again, he needs to choose.
- Not sure what I'd do about the OW's email. She's either desperate, manipulative, or both. I'd be inclined to answer in the heat of the moment "Fuck off. Don't ever dare contact me again." But I might think on it overnight, and then at that point, she becomes so unimportant she's not worth bothering about.
- YOU ARE AWESOME. Look in the mirror & say that whenever you doubt yourself.