I've not read the whole thread, OP, but I have read your posts and your previous thread.
I just wanted to pop on and say: you've done the right thing. People judging you and telling you things like 'you've given up' have no idea what it's like to live with somebody like this, even if it is your child. In fact when it's your child I'm sure it's worse, as it goes against your instincts to not want to live with them.
Anyway, we reached the point you have now when DS was a bit younger than your DD, he was approaching his teens. His behaviour was similar though, and the things you said about the atmosphere and constantly feeling on edge, waiting for the next thing to kick off, really resonate with me. It is no way to live, it made my whole family ill, but particularly me and DH.
DS lived with his father for just over a year, before his DF and DSM split up. largely due to realising that parenting him really was fucking hard work and that, actually, they couldn't do a better job than us despite their constant interference and undermining that made everything worse. He came back to us following their split as his DF was found, by SS after school made a referral, to be incapable of even the bare minimum of parenting by himself.
Since DS has come home, he's like a different child. He is now 16. He's a true delight to be with: funny, kind, hard-working, thoughtful, even if he is still cheeky, it's now just in good humour. That year made him realise which side his bread is buttered, that his DF was causing far more harm than good and that boundaries are actually good.
In the mean time, the rest of us had a year without the stress and managed to rebuild our own health and our relationships with DS.
It feels hard now, and deep down you probably feel the horrendous guilt and self-blame that I did. But this isn't your fault, and a child going to live with one parent isn't the failure of the other. It's simply a different parenting arrangement. Sometimes Dad's need to take their turn with the hard shit too.