I lived in a 15' camper for a year and a fifth wheel for four years, both in the same RV park as my home base. It was plugged in the entire time. Most of the 60+ other RVers there were plugged in for years. Nobody had any problems. It's done all the time. I would be gone for months at a time in my van and always left it plugged in to keep the batteries charged. I had a surge protector at the power pole (50 amp connection) just in case.
When I was "home", I checked battery water level every month or so and rarely had to add (distilled) water.
I think leaving it connected to shore power would be the better thing to do as long as you check on it now and then.
As for mice, yes, I did have a few in the underbelly, always in the fall. I cleaned everything with bleach water, stuffed every possible entry point with a metal shards infused foam, set traps and poison in the underbelly space (never caught anything). Ran rope lights all around on a photocell dusk to dawn thing. Had one wire chewed to the awning lights. Fixed. No big deal as long as you keep an eye on things.
I wouldn't leave a space heater on unattended. Plugged in with everything off (fridge, furnace etc) the power draw will be minimal. Just enough to keep battery charged. In your climate, I may be tempted to leave the furnace on a low setting to protect electronics (TV, etc)? Don't know about that.