Your personal liability depends on your state law regarding the family car doctrine, so the answer there is "maybe" (Texas is not a state with that doctrine, so simple ownership of the car does not confer liability). You would be liable if your supervision of the child was negligent, which means approximately that you knew or should have known that she was a bad driver and would cause damage. Still, the insurance company is suppose to indemnify you (plural) against such loses, as long as they are legally required to do so. That would include many considerations, such as whether the driver was covered under the policy, whether the car was being used contrary to the terms of the policy (being used commercially), and so on. The insurance company is entitled to make a reasonable determination of whether they are responsible for the loss (and if so, to what extent). If they actually decline to cover the loss, you would need to sue them to make them comply with the terms of the policy (and your lawyer would give you a detailed explanation of why they are not liable, in case they aren't).
The plaintiff works with his insurance company to recover his insured losses, and with his lawyer to recover any uninsured losses. His insurance company works with your insurance company, up to a point, and then the lawyers get involved. Your daughter does not work with his insurance company, and your insurance company probably has said something along the line "only talk to us".
The insurance that a driver typically has may cover some of their own medical costs, but does not provide a payment for "pain and suffering": that is an uninsured loss. It is not generally required that drivers carry insurance to cover their own medical expenses – it is required that they insure against damages, in general, suffered by other parties (if the defendant is at fault). So there is probably nothing for the plaintiff to work out with his insurance company.
In Texas, if the defendant is entirely at fault, defendant will be liable for 100% of plaintiff's damages. If defendant is 90% at fault, defendant will be liable for 90% of plaintiff's damages. If defendant is 49% at fault, defendant is not liable. Defendant can, in any event, also sue for damages, so if defendant is 49% at fault, defendant can recover 49% of her damages. The insurance companies might be able to talk it out and reach a clear resolution of the matter, but it could be more in their interest to throw the dice and work it out in court.
One can always sue at the very start, and drop the suit if it becomes advantageous.