Faced the same problem yesterday/today (FP2 Won't turn on or charge - See first post for troubleshooting). Phone run out of battery and shut down but it did not charge. I experienced the same rebooting loop as you do.
I had to charge the battery with an external charger. Now it works fine again. In contrast to you I did not face any problems with random shut downs before. I hand only some problems with my first slim case, as buttons got sucked.
Hope the external charger will also help you.