Can someone put your car on their insurance policy without your permission and without you being listed as a driver?

Who knows. Even if they do, you have no insurance coverage and they have wasted their money. The insurance company can not and will not pay any claims arising out of any incidents with a vehicle that does not belong to to the policyholder. Purchase your own insurance. It is a legally binding contract in which you state that you own the vehicle and that you have listed all household members and/or drivers and that you agree to notify the insurance carrier of any changes to such.