OP, I am an ex, and what I would say is keep an open mind:
Firstly, there is no such thing as "refusing to move out or give him equity," if he has a financial steak in the property then that will be decided as the consent order is drafted as part of the divorce. It's possible that your DP is bitter because she is refusing to discuss these things amicably and they are instead having to go to court which will be costing extra, but either way things will be decided. And if for any reason she is enabled to live in the house until the youngest is eighteen there will be valid reasons for that.
Secondly, have you seen these abusive emails she sends him? What is she emailing him about? Remember that the screen does not convey emotion, so a snippy email may not be intended to be that, alternatively it may be, in which case he just has to field the emails according to the ones he has to deal with, and the rest is just the ex being the ex, iyswim.
WRT the relationship with the DC, it is of course a good thing if you have a positive relationship with them. So many posts on here and elsewhere talk of the Sm's need to step back/not tread on toes/not be too involved, but to be honest I would be led predominantly by the children on this one, and if they want you involved in their lives then be involved. What I would say however is that it's understandable that the ex may have an issue with you if you were already living with her children's father at the point the children were introduced to you. That way it's happened almost as being a ready made family, and is a bit off really.
I remember when my DS met my eXH's then new partner, within weeks the ex was talking about planning a big family holiday with him, her and her DD who DS hadn't even met yet, while at the same time criticising me for spending time with my DP and DS, even though eXH had forced the introduction between DS and DP. I took a bit of a dim view of that TBH, given DS hadn't even met the DSD at that stage and suddenly they were all being branded a family. I only made my thoughts on that known once however, but I do know that since then eXH has painted me as being unreasonable, snippy, has gaslighted his DP into thinking I'm using opportunities to have a go at her when that is far from the truth. But unfortunately my DS hasn't taken to her at all and this has just fed into the notion that I must be the bitch ex (PS, I don't know for a fact I've been called a bitch, but I do know that she's been told I've had a go to him when I haven't.)
Why they are divorced is irrelevant here. Whether she had an affair or not doesn't matter. He's moved on, he's with you now, and the rest will take care of itself. You just need to let go of that element.