Re-starts by itself. Won't start. Loses 12V battery power?

Used to be occasional, now seems to be happening every other time.

When parking & pushing park button followed by pushing the power button to turn it off - it will appear to turn off. About 5 seconds later it will turn itself back on (or at least the MDF and dash lights appear to say it's on). Another push of the power button seems to finally really turn it off. When starting it back up the next morning the radio station pre-sets are all lost, the clock is set to 1:00, the trip mileages are set at 0. It appears as if the 12V had been disconnected. But there are no 12V battery related issues present after that or while driving. Has not displayed "12V disconnected systems" when driving yet.

Another thing that has also started happening recently is that at times it will not start when brake is depressed & the power button is pushed. After giving up for awhile and trying again a few minutes later, it will eventually start. Pretty sure this another symptom of the same problem causing the self starting symptom. So far it eventually does start after a few attempts but concerned it will eventually not start at all. Or that 12V power might be lost wile driving.

Any ideas of the problem and the fix for it?