Gap insurance will only cover what is left on your loan after an insurance company pays out. Gap coverage is best purchased if you are going to have a difference between the loan amount and the value of the vehicle.
A non-owner's policy will have a signed endorsement on it stating the insurer will not cover any claim that results from a vehicle the named insured owns or is registered to. Many will also state if it is part of the household, they will not cover it.
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.
If your parents are renters, they are not responsible for the property — just their own liability and possessions — so it's not required to have sinkhole insurance since that's property-related. The landlord has to maintain safe living conditions for their property.
LexisNexis is the company that maintains the insurance claim database. If you are disputing a claim, especially if you're the victim of identity theft and you did not file it, you should contact them about getting it removed.