I called our local distributor for Ruud, explained the whole thing to them, and they think it's a wiring issue. Referred me to a local contractor who is both HVAC and a licensed electrician. I have a call in to them, but can't get them to call me back.
Still haven't heard from my latest HVAC guy who has been doing all the most recent repairs. He was pretty upset when I called two hours after he'd left the last time to tell him it had blown another fuse...but I'm getting pretty aggravated at having a heat pump that doesn't work. We won't even go into the $350 electric bill from running emergency heat the majority of last month.
This makes me wonder if there hasn't been an issue with the wiring all along. It certainly would explain the wonky thermostat.