I can feel how catastrophic your current situation is. Something clearly needs to change here.
It would be easy for me to advise you to ask her to move out if you hadn't expressed that you don't want to separate your children. I agree with that sentiment. However, I don't think that this means *you and her* need to stay under the same roof. In fact, it sounds like the worst possible option.
I'm afraid I don't actually believe that building above the garage will help. Not unless you can all raise the cash to turn that into a completely separate living unit. Moving her bedroom above the garage will only offer her a bit more privacy. It won't change the fact that you are sharing a home. It won't change your relationship.
Have you considered selling your current house and buying one that could be turned into two completely separate living units? Or, of course, buying one that already has a separate living unit attached? Is splitting your current layout an option, to create two self-contained studios? This way, the boys can move between the two spaces safely and easily, but you two don't have to have much to do with each other.
If this is not something you want to do, and her moving out is not an option, it would be wise to seek family therapy. You could also do your own research and work on letting go of hatred. It's a heavy weight to carry around. I'm very spiritual, so Buddhism and mediation works very well for me personally. However, if she does have serious issues, you have to decide whether you want that influence in your orbit. Even if you 'Om' your way to personal tranquility, you have to consider whether you really even want her around.
This leads to my final piece of advice, and I hope you take it kindly, as I intend it. So often, we allow ourselves to believe we are trapped. All this does is increase hostility and negative emotion. In reality, there is always something we can do. Whether this means changing our mindset or making a tangible lifestyle change. You do have choices here, and it might help you to embrace that. For sure, sometimes it's a "lesser of two evils" kind of choice, but realising that you do have some control over your own life can be extremely empowering.
In your situation, this might mean that the lesser of two evils is that you have to make the sacrifice of living in a smaller space or cheaper area in order to keep your children together and escape sharing a communal area with her. Some could argue that this isn't fair on you. However, the way I see it, it's not about fairness - it's about finding a solution that keeps your children happy, and keeps you happy.
Me: (30f) open poly
GF: (40f) My long-term, long-distance partner
Hubby (37m): GF's husband
Garcon (26m): GF's submissive/third partner
“Peace comes from within. Do not seek it without." ~ Buddha