[Eugene]

My gaming PC is housed in a Bitfenix Shinobil XL, one of their largest cases.

I like the Prodigy, but it's not that small of a case. You linked the "mITX" version, which only accepts the smallest motherboards and has a difficult internal layout to work with. The "mATX" M version has the same external dimensions, but a more traditional layout. It also can mount mITX boards, so there's really no reason for anyone to buy the "mITX" specific version other than the marked down pricing.

Yes, HDMI out should work fine and the video quality is completely dependent on your TV.