Does it matter that doctors are billing my health insurance?

I was hit by an uninsured motorist and have been seeing my doctor and physical therapist for back and neck pain. Because of some initial confusion over whether or not the at fault driver was insured, I started out having my doctors just bill my health insurance like normal. I was also told be an adjuster with my auto insurance that it was okay to do that. But would it better to have the providers bill the auto insurance? I know that when it comes time to settle, my uninsured motorist coverage will pay those medical bills and I'll have to reimburse my health insurance, will the health insurance negotiate the amount, or not? Or if I have the bills sent to the auto insurance, they don't get paid until settlement, and then I pay them from the settlement, is that right, and I might get them to negotiate the amount? I'm just not sure what is the best thing to do, and as soon as we got the official word that the other driver was not insured, my auto insurer has stopped being helpful, at least as far as answering my questions.

Also, the initial repair estimate on my car was $2800. It has been in the shop for two weeks, and the amount has gone up to $5600. A friend of mine saw the car at the Chevy dealership yesterday, which is not where the body work is being done, so now I'm wondering if they found some mechanical problems too, which will increase the repair bill. Is it possible that the insurance will decide to total the car, even after authorizing the body shop to order parts and start work on it?