Your gap coverage should still cover the $2,000 owed on the loan regardless of what happens after that. I would definitely check directly with your insurance company to make sure there is no stipulation that prevents this if you choose to buy back the salvage title.
Given your claims history, I would pay for the damages to both vehicles out-of-pocket. You are correct that having any more claims added will likely make it very difficult to find coverage — not to mention when you do, it's going to cost even more than your current bills.
Unfortunately, you and the vehicle were not insured during the time of the accident, and if no policy was bound at that time, then you would not have coverage for the accident. Insurance will never backdate and trying to get them to cover an accident that already occurred is considered fraud.
It's hard to give a direct answer to your question, as restricted licenses are considered on a case-by-case basis. Some insurance companies will provide such insurance, but they too would have to know all of the details surrounding your situation.
Every state has separate laws, insurance plans, and registrations, but you shouldn't have an issue buying a vehicle in one state and moving to another state. If at all possible, I would wait and buy the vehicle in California — this way, you would only pay all the fees once.