A, if your h is willing to give you the house now without a future equity payment get it in writing and notorized. When my xh left he wanted to get rid of the house immediately. At that time there wasn't even enough equity to pay a realtor, but xh was willing to pay the difference just to move on. I landed a good paying job, we took the house off the market and I've been paying the mortgage ever since. There is now considerable equity and xh is fighting for half. Under the law is my state he is entitled to that as frustrating as it is. I sacrificed, he partied, but we are entitled to an equal split. Fortunately there are enough financial offsets that xh won't see a dime.
All of your communication regarding the house, etc. needs to be in writing. Document and keep everything. It may seem insignificant at the time, but you may need it later.
From your goal list you are clearly a survivor. I know this is terribly hard, but you can do this. Keep a vision of the life that you want and work toward that. You can't know how this will all work out, but just know that it will. You will move on and hopefully will have tremendous personal growth from your journey.