One month? I'm surprised it's that fast. No, really.
The statute of limitations for personal injury claims in PA is two years. Typically, sometime after the accident a claim is filed with insurance, who then (most of the time) pays it. The cases that don't get paid out right away linger, sometimes right up until the two year mark. THEN a case is filed.
Yes. As long as the claim is filed within the statute of limitation in your jurisdiction (Pennsylvania). It would be prudent to contact your insurance carrier as soon as possible to alert them of the claim (if they are not already aware) and they will handle the claim and assign an attorney in the event that the injured party files a lawsuit.
This advice should not be construed as forming an attorney-client relationship.