Will this ram work in this board?

No. the ram you selected is registered for server boards only. I recommend gskill ripjaws 8 gb 1333 for $109.99 at newegg. I use the 4 gb version of this ram in my system (16 gb total) and it runs fine. I got mine as a shellshocker special for $70.
 
Yes. Registered ram won't even post in many boards. It's tested rigorously against failure and designed for specific boards only. Some server boards won't even run with the wrong ram. The odds of it even posting are low, but if you don't believe me, write gigabyte or check their ram QVL listing for your board. You won't find any registered ram anywhere on the spec sheet. http://download.gigabyte.us/FileList/Memory/mb_memory_ga-a75-ud4h.pdf
 
Yes. The case should specify whether it supports an atx motherboard. Only a few old cases don't. If you can't find the spec sheet, the interior should support a board that's about 12 x 12 inches. A few old cases are btx format; they won't work with atx boards. The opening for the motherboard ports is on the wrong side.