What you're really asking for is a gaming PC that can do some HTPC stuff, like the TV tuner card and remote control as well.
Quiet and gaming don't usually go together often.
And it will be the gaming requirements (CPU/GPU/PSU/case) that drive the cost.
You'll be using your 1920x1080 resolution as your gaming monitor?
Is this the type of case you're looking for? SilverStone Lascala LC10
It's HTPC looking, but can fit full size gaming parts.