We have a 2015 36FK and while we haven't had the same things happen, we've had a few instances where I went to bed with the thermostat in gas mode set to ~64 degrees and woke up to find it had reset itself to 72. I've never seen it run with everything turned off, though.
The radio thing is interesting. I've accidentally left mine on before not remembering that it is powered by the coach 12v system and not the chassis. When I first fire mine up, it defaults to the nav screen. Maybe you had an intermittent 12v connection that resulted in it turning itself on?
The more complex these things become, the more opportunity for gremlins I guess.
Ty & Deb Montag
2015 Jayco Seneca 36FK
2014 Honda CR-V toad