I'm planning to build a fully passively cooled media/gaming PC to keep at home when I go off to 2nd year Uni, so at the end of Summer.
If it was me building the HTPC, it would be something like a low end Ivy Bridge, with a passive CPU cooler, then I'd look into getting the lowest volts possible for the standard clocks.
You can fit a standard ATX motherboard in that case you mentioned, so I'd be tempted to do that, if it was Ivy Bridge it would be a Z77 mobo.
For the GPU, if you need to play a demanding game, you might need one of the very quiet mid range GPUs, thinking cards with an MSi Twin Frozr cooler, or the Gainward Phantom cooler, then again underclocking it to get better temps, so perhaps a 560Ti but that might be more than you need?
Then maybe a small SSD for the OS along with a Western Digital Green drive for storage.
The fans in that case are quiet but you could probably upgrade?
Then I guess you'd need a soundcard if you love your multimedia?
Then that would be just about it.
The main thing would just be the heat issue, it is going to depend on what you want to play, if you need a low end card then there are passively cooled options, I suppose with the new AMD and nVidia cards coming out on 28nm with much lower temperatures we might see more passive cards on the market. It depends also on how much you care about noise.
In reality it's going to be easier just putting a new build in your current case