The radio can be turned on after the car has been off for 1/2 hour and without the key in the ignition?

Yes, electronics can go out without warning and no damage is done such as an accident. The Front Audio Control Module is what runs the radio controls and the seat belt warning chime. To be sure we should do a CAN scan which should show a stored code which will point to the FACM. You can get a CAN scanner (Controller Area Network) which will work on most cars from Amazon.

Here is a video to show you how:

https://youtu.be/u-4syLc-ifQ

and

https://www.2carpros.com/articles/can-scan-controller-area-network-easy

The FACM is located in the center of the dash. Here are the instructions on how to replace the unit in case a code comes up for it. Check out the images (below). Please let us know what happens.