VB - you're doing great, really proud of you. I know how hard this all is, I really do. Tax credits - I know he's still living there, but they will pay them from when you "separate". This can be living in the same house - but apart - i.e. separate sleeping arrangements, etc. They can also be backdated from when you separate, so perhaps from when you started divorce proceedings? You'll have to look into this. They can be quite sizeable, and because of this, they can affect legal aid - (if you have this for your divorce - I don't know how much you've paid or are paying - but your solicitor can advise you). I rang the tax credit office, rather than fill all the forms in, and they sorted them there and then. They were paid from the date he left, and I got my first backdated payment two weeks after I applied, and now they're paid monthly.
You've done the right thing about contact - of course he can't take the kids to spend time with OW; both he and her sound a bit unstable, and then there's her husband to consider. So whilst he should have access, it has to be on your terms for now, and that can be reviewed further down the line.
He can see them, as pickgo says, at a mutually agreed venue - again your solicitor needs to get involved here.
Detaching is hard, but essential. It will give you the upper hand, because you'll be able to get clarity of thought if you see all this from your perspective only. By focusing on you and the kids and your needs, things become clearer. Once you try to get inside his head, and try to work out what he's thinking, it all becomes cloudy and messy. Take this from one who's spent months doing this over the past year or so: you really can't rationalise an irrational person. And you're right - he's like a stranger to you. I was married for over 22 years, and my ex is like a stranger to me. So I just deal with the practicalities, and leave him with the mess he's caused. Think yourself lucky, you don't have to live with him much longer, but he has to live with himself and the aftermath of this, for the rest of his life.
You will move on, and you're already doing so: so keep posting and leaning on all the posters here who can help; focus on what's best for you and the kids - and leave him to it: you don't know yet how OW will feel about him when he's properly available, or about what her husband may do: so anticipate that things could well implode for him, but it's not your problem, it's his. You're moving on. He smashed his marriage into pieces - let him deal with this - on no account try and piece things together for his sake.
Good luck.