Unless you are the administrator or executor of your father's estate, you can not appear for your deceased father. A lawyer will be needed to represent your deceased father as the litigation proceeds. In all likelihood the Estate of your father will need to substitute for him in the litigation. Regarding the State not appearing, we would need to know more, including whether the State was properly served.
If the doctor or his staff did not submit the claim to your insurance company in a timely manner, and his untimely submission jeopardized your coverage, then the doctor should be estopped (prevented) from trying to collect from you directly. Your doctor's failure to follow the requirements of the insurance company resulted in you going without coverage. The doctor should not have submitted the bill for collection to his attorneys. Try explaining the situation to his lawyers and they may stop...
The injured party may be entitled to damages from the negligent driver for personal injuries or damages. The insured status of the negligent driver is not relevant for filing suit. However, an uninsured driver may not have any assets against which to collect a judgment for damages. That's where insurance provides payment for their insured's negligence. Without insurance a significant effort is made to obtain a judgment that may cannot be collected.
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Call your insurance professional and seek input for the appropriate coverage and policy. Often times a GCL (General Commercial Liability) policy excludes operation of vehicles and personal auto coverage excludes using a vehicle for commercial purposes. Your insurance agent or broker should be able to help. If you employees are in the scope of employment, liability will flow to you or your company.