If the machine is only doing backups, i.e. you have no requirements otherwise as to what applications should run on it, it shouldn't run as an HTPC, etc., you might want to look into using Solaris, a Solaris spin-off (IllumOS, OpenIndiana), or a *BSD variant, as they all support ZFS as native kernel modules (Linux also supports ZFS, but only as a FUSE module - it's slower, though very stable, I've been using it for almost two years in a 14TB array).
ZFS supports compression, snapshotting (enabling you to roll back to the time that any snapshot was taken), deduplication (discarding duplicate blocks to save space - but very very slow), and filesystem-level redundancy. The latter being the coolest of all, in that you can set up RAID arrays that are equivalent in redundancy to "normal" RAID solutions, e.g. you can have single, double or even triple parity on each blocks, but parity-checking and recovery is done on the file level (all files are checksummed and continually verified on-the-fly on reads), so you'll never experience silent corruption of data, as the FS is protected, not the array.
As far as cases go, it really depends on how many disks you plan on stuffing in there. I have a fantastic Lian Li case with 12 disk bays in a dedicated section at the bottom of the case. It's all-aluminum, very cool, and very quiet. The build quality is amazing. I recently modified it to hold an extra 6 drives by dismounting the door and removing all 5.25" bays. My particular model seems to be out of production, but they have quite a few really neat cases. Check out their collection of larger ones here:
About the hard disks, then no, I wouldn't focus on their speed so much. Using a lot of disks with parity, you'll soon run into bottlenecks elsewhere. In my particular setup, I have disks on an internal ICH10R controller (PCIe) and other on old PCI32 SATA expansion cards. The PCI bus is completely saturated when doing contiguous reads, but I still get 175 MB/s throughput. Plenty, plenty, plenty. Twice what I can expect from gigabit ethernet anyway.
The Western Digital green drives are somewhere in-between slow and fast, but the two 2TB ones I have are already showing a little too many relocated bad sectors for my liking, and they're under a year old, so I wouldn't personally recommend those. I personally just go for the cheapest until getting enough aversion against the brand to try something new. Currently, I'm not too fond of Samsung and Western Digital, but have a lot of very old Seagates (almost a decade) that are still 100%, so my last two 2TB drives were Seagate, too.
With regards to temperatures, keep in mind that the Google harddrive paper from a couple of years ago hinted at temperatures not being as important as previously thought. That is, when still within sane values, of course, but whether they're spinning at 20C or 50C it seems isn't as important after all.