I got a sidewall puncture and had to drive home on the stock donut spare. The car didn't sound happy, and I didn't drive it anywhere unnecessary, just home... (Approx 2 miles, city streets / hills) Then I got another WRX wheel and drove it to the tire shop the next day, where I got new tires put on. Now, the car is very VERY sluggish. It feels like something is binding. I've checked tire pressures, theyre at 38 front and 35 rear. It feels as if the brakes are binding, but they're not. Also, if I try to coast down a hill in neutral, the car seems like it's slowing itself down.

So I proceeded to do some not-very-technical tests...
I tried doing donuts in a gravel lot, it didn't want to kick around much. Also when doing a U turn at full throttle, it kindof Hops, and feels like the torque is going front to back very quickly, jerking back and fourth.

Also, I tried going up a lot of steep driveways / and offroading. The front seems to spin more, but under a light load the rear will still burn out 1 tire. I have a 20mm rear sway so the unloaded rear tire gets off the ground easily.

My car has 125k miles on it. How long does the viscous coupling typically last under normal conditions? What about severe conditions? When I got the 20mm rear sway, I had crappy all season tires and I was getting sideways everywehre under lift off. Will this bork my center diff?

Cliffsnotes:
got a flat, drove with mismatched sizes, may have messed up center diff. How can I test it?

Instead of getting a new / used center diff, could I just weld the center diff and have permanent 50/50 AWD? Would this present any problems?

Also, I wonder if the viscous coupling from a manual trans is similar in dimensions and gear teeth to the automatic electronic one 4EAT comes with.... Cheap DCCD?

 

This is the only question of yours that I think I can answer.

http://trucktrend.com/features/tech/163_0206_4x4/
"The connection between front and rear driveshafts in the transfer case is mechanically locked in part-time systems, which is why they can't be used on pavement where there is no tire slip. In full-time and AWD systems, the connection must have some means of allowing a small difference in front and rear driveshaft speeds so the truck can get around a corner without binding.

This can be accomplished by using another (center) differential in the transfer case, so front and rear drive can operate with some independence."

Here is a series of pages written for us laymen on why part-time 4WD cannot be used on pavement, with links to a pages explaining why full-time 4WD (and AWD) is okay on pavement, and what happens if there is nothing to provide a differential action between the front and rear axles.

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0==WW==0
"…axles of evil…" - george w. bush