If your daughter was the listed driver and the accident was logged correctly, most companies won't charge that against you. If you are looking to switch insurers and her accident comes up on your claims history, I would obtain proof or confirmation from that company that the claim wasn't yours so it can be cleared.
In short, a non-owners policy is for someone who doesn't drive a vehicle on a normal basis, doesn't own a vehicle, and doesn't have access to one. So if your daughter only rents cars, then non-owners insurance may be a better fit.
You should still be able to open up a claim for any injuries or medical visits resulting from an accident. Specifically, in Tennessee, you have three years to make property damage claims and one year for injury claims.
Yes, it would be a good idea to remove your daughter from your insurance policy due to her recent DUI and suspended license — both of which would impact your insurance by way of a much higher premium. Find out just how much a DUI can impact your rates here. Even with the citation and suspended license, she'll be able to find her own insurance — though that may be difficult.
If your daughter lives with you and you two share an address, your insurance company is probably assuming that she may sometimes drive your vehicle, at which point she would need to be a named driver on your policy. If she has her own car — and doesn't drive yours — it could be possible for you to have her excluded from your policy.
Unfortunately, insurance companies list almost all single-vehicle accidents as at-fault on your claims history. This would be considered an at-fault collision accident. There is no other party at fault and this can be seen as a failure to maintain control of the vehicle.
You should add your daughter and the new car to your policy if she'll eventually be driving it once she gets her license. This will prevent the insurance company from having any reason to deny a claim in the event of an accident.