You really need to consult with a personal injury attorney in your state. Your fact pattern reflects you have not accomplished anything on your own. Most accident cases are taken on contingency fee and most consults are free. So you have nothing to lose in contacting an attorney about your case. You have everything to lose by trying to work it yourself.
Given the limited amount of information you provided, it is difficult to give a comprehensive answer. I didn't see anything in your question about the amount of damages you sustained. If you did not have any bodily damages then the insurance company is providing a minimal amount to resolve any issues. The personal injury portion of these cases are determined by the amount of bodily injury you sustained. So from your question i can't tell if you sustained any personal injuries or not.
I agree with all of the other attorneys who have posted an answer to your question. A potential medical malpractice case is complicated and the records need to be reviewed to determine if there was any negligence by the health care provider. Although what you have stated is likely correct, there are other facts that factor into this equation and the only way to resolve it is to look at the records.
The answer to your question is, it depends and I don't feel there are enough facts here to assign fault. If he negligently accelerated then quite possibly the fault could be assigned to him. The insurance company will either accept the fault or they will dispute it. If they dispute it, they will inform you as to why they feel their driver was not at fault.
Sounds like you may be having an allergic reaction. You should go and see your doctor to determine if this is an allergy. Even if it is not an allergy if the symptoms are worsening, you need to see your doctor.
I agree with the previous attorney's post. There is more to a claim than just a demand letter. And attempting to piece together legal services will only prove devastating to your claim. You need to consult with an attorney for the personal injury claim, for the entire claim, not just bits and pieces.
What proof do you have as to who impacted your car? Even to file a small claims case you will still need to show your proof. Evidence could include photographs of the other persons car that impacted yours with the paint still showing on their vehicle. Other evidence could include any eyewitnesses. With the cost of litigation, small claims may be your better route if you can come up with the proof you need to prevail in your claim. Good luck.