To build a RAID system, all HDD must be identical, i.e. same capacity and same speed, not necessarily the same brand. However, different brands usually have slight differences in performance too, so it is advised to use identical drives of the same brand too.
Rack cases are different from “consumer” cases, and probably you’ll be forced to buy a PSU purposely built to be installed in a rack. These PSU are very costly, and not that easy to find.
In my opinion you can spend a lot less money buying a NAS. A two bays NAS and a couple of 3 TB drives should be more than enough to store a rather large collection of movies and music
However, if the rack case can contain a regular ATX board, you’ll need a board with as much SATA ports as possible. And these are very costly too.
If the server will be used as media storage only, you can save a lot of money getting a low power CPU. An Ivy Bridge processor is a good choice, because is powerful enough to run a server and it has also a low power consumption.
A low power consumption means also that you can use a silent CPU cooler. Theoretically, you can also use a large aftermarket cooler designed for overclockers and use it without any fan, so the CPU can be cooled passively. These coolers, however, are quite costly too, so again the NAS seems the best solution.
Because of you said to have different disks with different capacities, I again suggest to get a box like the one I suggested before i.e. this or the 4 bays version.
With that box, you can use all HDDs as single drives. You can remove or add drives as you like without any worry about rebuild the RAID array, just power off the box, change the drives, and power on the box again.