I think the symptoms you are having with the heating system are consistent with the cause being a loose connection in the thermostat wiring. Since you can just hang the thermostat by the wiring and make it work, I'd start by checking the connections of the thermostat to the thermostat wiring for the heater.
There's lots of advice and information in forums... sometimes it is correct. For example, all of my posts are made by a political appointee who got the job as a reward for contributions to my diesel bill.
2011 Jayco 28.5RLS; 2008 Chevy Duramax; Pullrite Superglide Hitch