Will this Gaming Rig Work?

Yes, definitely choose that PSU. It 1/2 price off and it's 2nd most bought PSU on Newegg. I just bought this today. Just like you, I was going to buy a cheap PSU but someone here told me the cheap one may fry your motherboard and anything connected to your PSU. So in the long run, $60 is cheaper than replacing a mother board and such.
 
probably, if you want to guarentee it then the 6950 2gb is about 260 or 270, and it can max out every game at 1080p or like everything but like aa or af, on of those 2 takes a lot of vram and power to do.
 
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