Companies can have different rules about who needs to be listed on the policy. Most commonly, anyone who lives in your household (that is of driving age), anyone that will drive the vehicle regularly, and any spouses of listed drivers will need to be added to the policy.
While it can vary based on your policy, your insurance company, and the weight of your U-Haul, many insurance policies do extend to rental vehicles. Again, while it can vary by policy, my experience is that if the towing capacity of your rental is less than 1-ton of less, your car insurance coverage will extend to your u-haul.
Most driving records insurance companies run will show the claim paid under "no fault" or "not-at-fault" and should not affect your rates at all. If your current car insurance quote reflects the accident as "at-fault," present the evidence you have in your favor and get a new quote.
Most companies will only run the current MVR so in your case Texas would be pulled and the Illinois ticket will not show up. The issue would be if the underwriters pull the Illinois records and see the ticket they will add it to your policy after you have bound it.
Although you're required to carry at least liability insurance in California, your uninsured status shouldn't affect your ability to have a claim paid out. In terms of what you will receive for your vehicle, you will most likely be cut a check for the actual cash value of the vehicle.
I recommend the following: See our guide here: Car Insurance for 19-Year-Olds : In our case study, Geico was the cheapest insurance for teens. Within this guide, you can find some other saving alternatives. If your daughter is in school, see if she qualifies for a good student discount.
Most, if not all, insurance companies require that any household members of driving age be listed on your policy. In most states, however, you have the option to exclude household members so they don't impact your rate and are not insured under your policy.