About 20 mins later, the "check engine" light came on, and the car started to die periodically. I parked it at taphophile's house and got most of the water out of the back seat. I checked for any obvious oil leaks under the hood. I checked the oil, and it looks fine. I checked the (stock) air filter to see if it was wet. It was moist, but not soaked. After sitting for about 2 hours, I restarted it and the check engine light was still on.
After seeking advice from some very helpful subaru techs, I decided to limp it home and clean it up there. I used the wet/dry vac to suck up any lingering water in the back seat floor. Based on their advice, I replaced the air filter and rebooted the ECU (the car's computer). The "check engine" light went away (thankfully), but I'm still going to have the differential and transmission fluids replaced tomorrow before doing any more driving. I'll have the technician give it a once-over to see if there's anything else that can/should be done for it. For now, it looks as though I got off really cheap.
Interestingly, the wrx's onboard computer is located under the passenger seat. I find it odd that the back seats flooded, but the front stayed relatively dry. I have a bunch of questions for the subaru tech tomorrow. For now, I'm just happy that it appears to be working and the damage (wet carpet) seems only cosmetic.
As a side note, working on a wet car in the rain sucks. Getting all dressed up to go shopping at a mall (with underwear, even) makes for an extra uncomfortable experience.