[BonzoHansen]

1) I hear about 8-, 9-, 10-, 12-bolt rears, but I have no idea what those actually are. Can anyone explain what they are, and the differences?

Off the top of my head...in Chevy-ese, the "bolt" count usually refers to the #of bolts in the ring gear.

Never heard of an 8-bolt.

9-bolt - The 9 bolt was built by Borg Warner in Australian. Used in 3rd gens. Considered stronger than the more common 10-bolt rear found in those cars. Includes positive axle retention (no clips). Although I am not sure if the ring has 9 bolts or just the cover. Matters not. Parts are scarce, IIRC biggest issue is the "posi" units wear out.

10 bolt - there are a number of rears that fall under this category. All are "c-clip" design, which can become a weak point especially as axles wear. When these break the axle comes out, hilarity & suffering ensue.

F-Body related there are 3 10-bolts:
8.2" - found in 1st gens, not known for its strength.
8.5" - found in 2nd gens, can be built fairly strong, stronger than most people realize. This rear has been in a lot of vehicles in some form or another.
7.625" - found in 3rd and 4th gen f-bodies, although the housings are not the same. This same basic rear is found in g-bodies and s-10 trucks and is a weakling. I always refer to it as a 7.5" rear. Or a POS.

12-bolt - found in 1st gens and a few 2nd gens (1970 only) - considered very strong and can be built to handle high HP. You can now get aftermarket versions to fit all f-bodies

Other f-body rears of note:

Dana 44 - IIRC there was a service package (parts dept only) replacement rear for 3rd gens based on the dana 44 design. I think Justin has one, they are probably stronger than 9-bolts. I've never seen one. I think it includes positive axle retention (no clips)

Dana 60 - a super strong & heavy aftermarket rear based on the Dana 60 design. Includes positive axle retention (no clips)

Ford 9" - a super strong & heavy aftermarket rear based on the Ford 9" design. Includes positive axle retention (no clips)

6) In general, where is the best place (online and off) to get parts for my F-body; aftermarket or "original replacement parts"?

that answer varies wildly based on the parts & user requirement.

8) What would be the best performance increase, in terms of cost-to-power ratios (the cheapest method for HP or Torque boost)? This can be specifically for my 5.0 HO TBI manual, or just in general.
I agree with Matt - the basics & rear gears.

Edit: Also suspension. Subframe connectors, better rear control arms, better springs/shocks & good tires make a big difference in those cars.