Center diffs typically fail from owner abuse and neglect. Poor tire pressures, doughnuts, hard launches, high mileage fluid changes, etc. The bottom line is: overheating and aggressive fluid breakdown lead to increased wear on the components.
Which axle did you put the spare on? If it was one of the rears, that very well could have caused serious issues as you mentioned.
From my recollection, with the car in the air and in netural, the front wheels will spin opposite directions. You shouldn't be able to move the front with the rear wheels on the ground. I'd double check this, but I believe this is a correct statement.
Best test is to take a highway drive for about 20 minutes to get the fluid to temperature, then go into a parking lot and drive in figure 8s at low speed, near full-lock. If the wheels hop/drag or the car is very difficult to turn, you have a bad center diff.
The braking sensation may actually be your brakes. Check if one of your calipers is sticking.
I'd recommend replacing the center diff with the factory unit. A brand new one can be had for around $450 from Heuberger (a forum vendor/sponsor). If you are mechanically inclined, you can do this in your driveway/garage with one set of ramps and some basic tools. It does not require a transmission removal. Here is a link
of photos to help you determine if this is something worth doing.