You will lose this discount and most likely your rates will be impacted if you file a claim. How much your rates will be impacted depends on the types of coverage you have and what coverage pays out to fix your car.
Firstly, a police report is not always necessary to file a claim, especially for incidents that only involve one vehicle. As for your second question, should you file a claim, the choice will ultimately be up to you.
Unfortunately stop signs in parking lots are not the same as stop signs on the street when it comes to determining fault. Parking lot accidents are often difficult to assign fault for insurance companies so they may ultimately decide to cover only their own customer's damage by saying the that fault was 50/50 between the drivers.
Regardless of the amount of damage caused in an accident, drivers should still exchange insurance information. Normally in the event that the police are called and a report is written the information of both drivers is documented, including insurance information.
In California, your insurance status does not have anything to do with fault after an accident. While it is technically illegal to operate an uninsured vehicle on public roads, fault is still determined for insurance purposes.
She really has the option to go either direction and her rate probably wouldn't be impacted since the other driver was at fault. If the other driver wants to pay for the repairs out of their own pocket then should wouldn't need to file a claim.
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.
in the state of Arizona, you technically have up to 2 years from the date of the incident to file a claim. While I would not recommend waiting that long, the point is that you are still well within the window to have your insurance cover the other vehicle's damage.
If you are having trouble getting the other driver's insurance company to pay out then I would definitely recommend contacting your insurance provider. If that doesn't lead anywhere then your only other option would be to contact an attorney.