If the charger is going to be connected to the battery and plugged in continuously, then it's a good idea for it to be one that can switch itself on and off. Some people call those "maintenance chargers". I have a 2/10/50A charger, and I'll use it in the winter on the low setting for a day or so to bring up the battery on vehicle that I'm not using, but I wouldn't leave that charger on continuously all the time.

If someone is going to get in there and start the car periodically, then yes I would have them run the AC as well. And when they run the engine, I'd suggest a good 20 minutes or so to dry things out. A running engine generates moisture, and in a cold engine some of that moisture goes into the crankcase oil, for awhile. It's important to get that case hot enough so that the oil drys out. And probably not a bad idea to put some fuel stabilizer in the gas tank as well, since that gas is going to be there for awhile.

The other one that crosses my mind (more of a northern car thing) is whether you have tires that bleed down air. Steel wheels are pretty good, but sometimes cast aluminum rims can leak at the bead because of salt corrosion. It sort of is what it is, and in this case there probably isn't a lot you can do about it except maybe have a small portable compressed air tank when you return. Farmers sometimes use these small tanks (say 7gal) to lug air from the shop to the field.