There is no way that you are going to find a card for that price that will play modern games on high, especially not BF3. You could probably play on medium though with something like the gtx 460. $100 is a good price. GTS 450 vs GTX 460
Lol there is no question about the motherboard, more the case. But your case is a mid tower, so you'll be fine for the 6850. My friend can almost play BF3 at ultra in 1080p with a 6850. At such a low resolution your processor is going to be the bottleneck. At that price point I would highly recommend getting a sandy bridge pentium and cheap motherboard to go with it. A couple more bucks for a lot more power coupled with a 6850.