1. Do I need to "warm up the engine" after "stationing" the 4Runner for the weekend? Seems legit with battery concerns, but I'm wondering of the stock battery is heavy duty enough to forego this or if there are other technical reasons for doing so.
"Warming up" the engine probably wouldn't hurt, especially if it has been sitting for a few days in cold temps (think below 20 F). I don't think its necessary, but necessary and preferred are two different subjects. I mostly turn my engine on a little in advance just to get the heater and defroster going.
2. Is an air compressor really necessary for a "snow kit" like the 4th Gens suggest? Do I need anything other than a shovel with my drivetrain
Not sure why you would "need" an air compressor due to the snow. It's not a bad tool to have around to modulate your tire pressure, but it's certainly not "needed."
A shovel isn't a bad idea, in case you get stuck. Having an ice scraper and brush is also recommended so you can clear out your windows prior to driving.
3. I consider myself a pretty reasonable driver in terms of safety, but is there anything I should watch out for from other drivers to keep myself safe? People talk about how no one knows how to drive in the snow, but I'm assuming that's just standard whining.
Some people don't know how to drive in the snow and there are accidents because of that. The key to snow driving is really simple: don't go as fast you normally would. Take it slow; give yourself a bit more standoff from other cars, especially when braking; cut the throttle if you feel the rear end coming loose. If you show even the slightest amount of caution and common sense, you'll be fine.
The reason some people "suck" at snow driving is because they barrel around corners and make split-second lane changes when there is snow and ice on the road; and then they're surprised when they spin out and end up on the road side.
I'm probably overthinking things and I'm really more worried for the wife who is from CA and drives a Honda Fit so any general advice that I can also pass along is much appreciated. And no, she's not getting my keys or a 4Runner of her own!
Does the Fit come with AWD? AWD with a good set of snow tires should be just fine.
Your limited's AWD, with the locking center diff, should also be more than adequate for dealing with snow and ice. Snow tires would make your 4runner even more capable, but with your lockable center diff, a set of A/T Tires should get you through most situations just fine.