Well, since I'm board, I think I will answer fairly thoroughly.
1. Thermal compound usually comes with the stock heatsinks that come with CPUs. Its usually pre-applied to the heatsink. It is needed for a heatsink to make a contact to the heat source(CPU, GPU, etc.) which fills in the really small gaps in the metal heatsinks. The downfall to using generic thermal compound to others like AC5 or Arctic Cooler MX-2 is that you don't get a very good thermally conductive contact. If your not going to be overclocking your CPU then using the generic stuff on the CPU Heatsink would suit you fine. If you are, then a aftermarket cooler and some aftermarket thermal compound would be needed.
As for a case, there all one the same standard, just pick one of these cases(Full or Mid Tower would be recommended, Full is around 2 feet tall and wide while Mid is about 15" or 16" inches tall). When you go to mount the motherboard, there should be little raisers that screw into the case that raise where the motherboard will be screwed into. I put the motherboard in first, remember the holes there are for mounting, and put the raisers in those spots.
As for a decent gaming computer under $1200, how does around 1000 or 900 sound? You honestly don't need $1000 dollars for a good gaming computer now-a-days. Here's the links from Newegg.com which is a good site for computer parts. I will have added the shipping cost, just got to add the taxes. This is also assuming that you have a current monitor that you wish to use.
CPU: Intel Core 2 Duo E8400 Wolfdale
Motherboard: ASUS P5E Deluxe LGA 775 Intel X48 ATX
Ram: CORSAIR Dominator 2GB (2 x 1GB) 240-Pin DDR2 SDRAM DDR2 1066
PSU: CORSAIR CMPSU-620HX 620W (Comes with Ram Combo)
Hard Drive: Western Digital Caviar 500GB 7200 RPM 16MB Cache
Video Card: Sapphire Radeon HD 4870 512MB
Case: Depends what you chose, so I'm guessing around $60 Bucks.
I am assuming you have a DVD drive and Vista, if not add around $250 for the DVD drive and a "RETAIL" version of Vista. If you get a OEM version then you can only install the OS once, no reformatting and such.
All of the above when added together AND a fee of .07% tax is added on and $10 of shipping cost, its around $1,082.15.
Just ask if you have any questions.