Keeping Her Plugged In

My coach is always connected to shore power unless driving down the road. No issues at all and when I sold my last coach the house batteries were 8 years old and still doing fine.

If you stay plugged in, check the battery water level monthly. Top off with distilled water as needed.

There is a small chance of an electrical surge causing damage.

Those are the only negatives.

I think it's good to keep enough heat going to keep things dried out, as in a small electric space heater. Also leave some vents cracked for air circulation.

If you are in and out, it's easier to watch for developing issues, and also is a deterrent to critters.

Unless you have an old style converter/charger that boils batteries dry, it's better to leave it plugged in and the battery electrically active. Slowly discharging is a recipe for sulfation to build up.
Many three stage chargers also have desulfating cycles as well - further ensuring the best life from your battery.

Why Don??

Better off to unplug from shore power. Just leave the cord beside the shore power plug.

Mine stays plugged in all the time with the battery isolation switches left on. The converter(if a smart charger) will keep the batteries maintained. Your "tinkering, repairing, adding stuff and general visiting will keep your whole unit better maintained than those that are ignored. It will cause you to "exercise" the stuff in yours more than you may think. Mine is about five feet from the back door in a carport and I'm constantly giving the old girl a lot of TLC. This causes me to really feel for the folks that have to store theirs.

As long as your on-board charging system does not over-charge your house batteries, why not? If you go camping long term, you'd have it plugged it wouldn't you? Occasionally check the house battery to make sure its not boiling over or over charging. If not, your good.

Yes.