Is the questioner sinful for his father going to the traffic department instead of him to finalize the procedures for a minor car accident he caused, especially since he does not have a driver's license, and what should he do after the report is written in his father's name?

What your father did by claiming to be the driver is not permissible, as it is a lie and forgery, and he must repent from it. Ignorance of the prohibition removes the sin, but not the liability. Therefore, both of you are obligated to return what the insurance company paid out for the accident. If the insurance was commercial, then anything exceeding the amount insured for third-party liability should be returned. There is no obligation to inform the police of the truth of the matter, and the owner of the damaged car has no right against you after receiving compensation from the insurance company.