Most people understand that if they were at-fault in a car crash then they will likely see an increase to their rates. Unfortunately, not-at-fault accidents can also affect the rate that you pay for car insurance.
It isn't too late to file a claim, even if you and the other driver agreed not to, since your insurance policy is intended to cover you after an accident. If you are considered to be at-fault in the accident then your company should pay for the damage caused to the other driver.
Normally insurance coverage follows the vehicle being driven even if it is borrowed by another driver that has their own policy. In a situation where you are hit by an uninsured driver and the vehicle you were driving is also uninsured, it may be possible to file a damage claim with your insurance provider since your policy would normally apply on a secondary basis.
If the two of you can't come to an agreement on the dollar amount then she is well within her right to file a claim for the damage. If you do decide to pay out of pocket I would highly recommend documenting the transaction somehow; either through a receipt, written check, or photo of the transaction.
If your son hit a deer and that is what caused him to go into the ditch then the accident is likely to be considered a comprehensive claim rather than a collision. Comprehensive claims generally don't impact rates as severely, if at all, the way a collision claim would if he had hit another driver.
You are right to be concerned about the car being totaled if you file an insurance claim, so if you can afford to pay for the repairs out of pocket then I would recommend doing that. If you were to file a claim, the car's owner might have the option to purchase is back from the insurance company but that could easily be avoided by simply covering the repairs yourself.
Since it was a parking lot incident, fault can be difficult to determine. If you two aren't able to come to an agreement on the repairs, your best bet would be to just contact your insurance company and let them know about the what happened.
Your best bet is to contact the police and file a report. If you were able to write down or remember their license plate number then that should aide the police in finding the responsible party. In the meantime, you should be able to file a claim with your insurance provider if your policy includes collision coverage.
If your policy was active at the time of the accident then you should be covered. While the insurance company may be a little suspicious at the timing of your claim with regard to it recently being reinstated, you should still be covered assuming that there was no fraudulent activity.
In Michigan, each driver is responsible for damage to their vehicle so the person who was hit would need to file a claim with their own insurance company. Unless your vehicle is also damaged and you want it repaired, you wouldn't need to contact your insurance company as the other driver would have to file a claim with their company anyway.