Its not always easy but any building can be mouse proofed. For years, we'd catch multiple mice in traps inside every week and couldn't figure out how they were getting in, but then inspired by this video, I built a mouse house with cotton balls, bait, and $10 of fluorescent powder. Sure enough, within a week there was a trail of fluorescent powder that could be tracked with a UV flashlight down a door frame, across the floor, and over to very small gap between the carpet and the wall where the sub floor wasn't cut tight to the log walls. After filling this gap there were no mice for while, but after a few months they found a new path. Repeated the process, and tracked them to a gap around the water heater vent: they were actually climbing up through the channels of a metal roof, and then down along the vent. After blocking that off, we have been mouse free for 3 years.

This is a log home surrounded by woods more than a quarter mile from the nearest building. If it can be mouse proofed, anything can. The only 100% foolproof method is to fully seal your house with no gaps a mouse can fit though.

If its any consolation, you can see on a satellite view (google maps for example) that the entire project area has had extensive logging within the past few years, so at least its not old growth or even mature second growth forest they will be clearing.

Eh, as long as its only a little it couldn't burn, but you should probably replace the regulator soon. They only leak when they wear out, but keep regulating for a long time after so people often don't bother.

I wouldn't, if the thermostats are calling for heat, it will keep the circulation going. Unless you are certain you have enough antifreeze in the system, there is a chance stopping the circulation could lead to a burst pipe in a cold spot.

The first place to be affected by the cold would be the tank regulator if it is outside. If you can stand it, it might be worth trying to warm it up with a hairdryer, as long as you don't smell any leaking.