Numbchux

Assuming it's just the clutches that are burnt up, no, it won't hurt anything. If it cooked the bearings back there, it's possible to have an issue. Yes, the driveshaft output and center "diff" housing uses the same ATF as the transmission. Front differential/front axle outputs has it's own gear oil.

Transfer clutches in the back of the transmission. These wear out under normal use, so I'm sure they're cooked. Pretty well documented how to replace them, can be done in the car.

Bad connection somewhere. Like these guys said, could be in the circuit up to the ignition switch, a heavy duty relay would take the load off of that, or the contacts in the starter are wearing (or both, one can cause the other).

5 speed? 4-speed? Factory tire size? Other modifications? IIRC, low 20s was about as good as I ever saw in my 4EAT AWD '6. So you're not terribly low for city driving. If there aren't any other symptoms, I wouldn't mess with it. Maybe take a look at the coolant temp sensor, those can get corroded and nasty and read incorrectly, which would cause it to run rich.

Probably best to be familiar with the diagnostic mode of the Auto A/C, so some tests can be run when it's acting up.

Feel the heater hoses. Hot upper hoses and cold bottom means the thermostat is almost certainly working (at the very least, there isn't any flow through the radiator, but if the coolant level was low or had bigger circulation problems, the engine would likely be hot and the upper hose wouldn't). If one heater hose is hot, there isn't flow there, likely core is plugged. If they're both hot, then it's an issue with the control (air is being diverted around the core, it's not calling for heat). There's a diagnostic procedure for the auto temp control in the FSM, I haven't had to mess with it on our '00-'04s, and I avoid the 05+

Yep, that's the tone wheel for the crank angle sensor. It's junk now.

Well, the exhaust and port is similar to the EZ30, but beyond that, the head has nothing in common.

Very interesting pictures. It looks as if the head casting might be the same, and just the ports machined out differently.

You're forgetting the '04-05 EJ259. Single port exhaust similar to an EZ30. AFAIK, the only thing unique about the 259 is intake and sensor related. The Block and probably heads are the same.

AWIC leakage is pretty rare, certainly on anything with any quality. But, they are a much more complicated system, and getting core/exchanger/pump all working together is kind of a challenge. I highly doubt you're making enough heat on an EA82t to justify all that. Just a simple early WRX one would be perfect.

Needs a whole control arm? If you're desperate, the dealer will be the most accurate way. They come with the ball joint, where aftermarket typically does not, but they can usually be had in a day or 2, and you'll be completely sure it's correct (looked up by VIN).

For the record, I took "back up" to mean reverse. But this is a much larger circuit, providing constant power to many circuits, and I don't think any of them are in the rear hatch.

Back up lights are in the hatch, right? These are notorious for having damaged wiring between the body and the rear gate. The dealership where I worked stocked rear gate harnesses (MSRP was about $85, I think there are 3 different part numbers, use your VIN to look up). Obviously individual wires can be repaired, but this will not be the first time you have it happen.

Oh yea, same idea on basically any vehicle with a return-style fuel system. Returnless cars just have a spring, no vacuum/boost reference (as the regulator is located in the tank, not practical to run a vacuum hose back there).