The Radeon HD7850 only requires a "500-watt" power supply if it's one of those cheap Chinese models that lies about its capacity. A decent power supply with 30 or more amperes of current available on the +12 volt header isn't going to complain. The Silverstone power supply in the Dorm PC 2.0 build can provide 36 A at +12 V. No worries.
Note that you could spend a bit less money, have much more expandability (4 PCIe slots vs. 1) and have many more alternatives available if you went with a Micro-ATX build rather than Mini-ITX.
i7-4770K, H70, Gryphon Z87, 16 GiB, R9-290, SSD, 2 HD, Blu-ray, SB ZX, TJ08-E, SS-660XP², 3007WFP+2001FP, RK-9000BR, MX518