Only a knowledgeable workman doing an on-site inspection can determine if your observations are correct. The arrangement that you believe you have is not at all conventional. Sounds as if it might not be to code.

Well, the house has presumably run like this for 25 years, so I wouldn't panic. However, that is not how you should wire a baseboard heater. It should be on a double pole breaker. That way, if anything goes wrong, both legs trip at once. In the current design, you've lost some of the protection that a circuit breaker is supposed to provide. You should upgrade the wiring as you are able to address this safety issue.

As for your electronics, you've got visual evidence that these circuits are not providing a consistent voltage to power anything on the dual use leg. You've also got some potential neutral/grounding issues depending on how the circuits are joined at the heaters. It is definitely a less than ideal operating environment for expensive electronics. They sell all sorts of surge protection/voltage regulating devices for electronics, but good ones aren't cheap.

I don't believe that you "... needed to shut off several seemingly unrelated circuits to completely shut off power to the heater and to the thermostat..." Your next sentence made it look like you needed to shut two.

Do a little more homework and post again. If you don't map out what is on which breaker, who will?

Thanks for the responses.

I think I'll call in my electrician and see if he can tell me what's going on.

It sounds a bit dodgy...