I am just wondering if you could give me some advices. At first I want a pc which can play some games and should not cost more than 800 $.
And this pc should be silent. I dont want to hear anything from it or only in gaming mode.
I thought at the CAse as Da Box 100 Blackheart
and for the Mobo : Gigabyte GA-H55M-USB3
As you'll need some fans, and have no space for a standard fan controller, I'd rather to go with an ASUS or MSI (as they usually offer better fan management). However, I guess your budget is a bit too tight, if you really want to cope gaming, small size (even if an odd size), and quietness: just as an example, the best fan management in BIOS you will get only with high end mobos, which (if in case) will drain lots of money.
As PSU due to the particular placing in the DA BOX (in front of you)
a Seasonic X-460 is highly recommended, IMO. So we can say 160 USD ?
Then, as GPU, I assume you won't go SLI/CFX: so the most quiet tradeoff I think it should be something like a Radeon HD6850/6870 with an AC Xtreme (you will need VRM and ram cooling also). At least 270 USD?
So I think that still without CPU, RAM, any storage, media drive, heatsinks, fans, other accessories and shipping charges, you may have already nearly exceeded (taking into account: 190 USD for the case and 110 USD for the proposed mobo, or 160 for some of the suggested ones) the aforementioned budget.