Yes. Your ex-wife can insure the vehicle as long as you are listed on the policy as a driver (because you are the one financing the vehicle) and listed as the additional interest since you have a financial stake in the vehicle.
Since your friend was driving your car for an extended period of time, they would need to be listed as a driver on the policy. If they are not listed on the policy and something happens, the claim can be denied for not listing all drivers of the car.
Congratulations! Paying off your car is a huge accomplishment. 1. Yes, let your car insurance company know: It is a good idea to notify your car insurance company of the loan payoff so that you can remove the lienholder from your policy.
Here's what I recommend: because you don't have a drivers license, don't have a driving record, and won't be using the vehicle, I would have your son get car insurance for the vehicle but I would use your bank/credit card information to pay for it. Most insurance companies will allow you to add another consenting individual's payment information to the insurance policy - as they want to be paid for the policy regardless.
To my knowledge, no car insurance companies use ownership of the insured vehicle as a rating factor. The only way your premium would be impacted by having car loan is if the bank or dealership requires you carry additional coverage you wouldn't carry yourself for a vehicle you own.