No, thankfully my STBXH and I are no longer living together.
I'll add a couple practical thoughts. If not already in effect, try to have your residence declared "in your possession". With his claims and entitled conflict, perhaps confrontations too, having that recognized will reduce the risk of him coming over and coming in anytime he feels like it.
You oldest may already be in school or soon will be in a few months. From the start, have yourself assigned or declared as Primary Parent, including for school registrations and contact. You'll save yourself a lot of grief by getting that cleared up at the earliest opportunity.
In my case, I was assigned the non-Primary Parent role in my 2 years of temp orders. So when our preschooler started kindergarten, she registered him in her nearby school district. We settled for Shard Parenting and equal time when I arrived at court on Trial Day but my single condition (firm boundary) to settle was that I be the Residential Parent for School Purposes. She begged otherwise but she knew a trial would not go well for her, considering her misadventures during the divorce process and a Custody Evaluation not in her favor. Both lawyers insisted there was no benefit being in charge of school. Were they so wrong! Less than 3 months remained of his school year and the school agreed he could remain until then. In late April they notified me I had one day to register him in my own school district, she had acted up one too many times. Object lesson: They would never have kicked him out (actually, mother) if she had continued as primary parent at school. They would have been stuck with her.
My ex has been constantly threatening to leave the kids (5 and 1) because he says I believe he is a terrible father and they would be better off without him. I have mixed feelings on how to reply to that, since part of me thinks it might actually be true. They might be better off without him.
Unless your stbExH is found to be a real risk to the children — substantive child abuse, neglect or endangerment — he is unlikely to lose his status as Parent. Courts really try to avoid taking such drastic action. While you might get temp custody at first until a final decree, the typical outcome is joint custody. However, if enough serious issues are documented you could be granted sole or full custody.
Beware of the feeling you have to appear overly-fair or overly-nice. (Observation: The parent behaving poorly seldom gets consequences and the parent behaving well seldom gets credit.) Remind yourself that your children's best interests (and yours too!) come first. In cases like ours — where our good efforts aren't reciprocated — we can't afford the luxury being overly-fair, overly-nice or overly generous. Just do what you have to do.
Do you have any witnesses or recordings documenting him stating the kids are better off without him or him abandoning them? Courts aren't impressed by vague claims "he always..." or "she always..." and often discount it as hearsay. So if you haven't started a log, diary or journal yet, then do so now to record the
details of incidents. The details (dates, locations, witnesses, etc) make it much more credible.