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.
In the case of a hit-and-run, if you filed a police report and have uninsured motorist coverage, this could be used to cover the damages. This is additional coverage that must already be on your policy.
As a general rule, any claim that is filed with your auto insurance company can increase your rates, even if it's just medical payments. If you are interested in finding out more about Med Pay, have a look at our article for more information.