Shivam15 :
yeah i didn't get the crossfire part. What exactly does it do?
Also i would like a motherboard which has uefi bios because in future i do intend on dual booting with steam os .
and if the extra gig doesnt matter than why do some games require 1.25 gb memory graphics card>?
Crossfire is the ability to use 2 GPUs (same model) together essentially as one. For example, you could buy two 7770 HDs and run them with a bridge (a connector) which will pretty much double the performance of a single 7770 HD. So you could buy one 7770 now and then down the road buy another. Nvidia has the same thing expect they call theirs SLI. To tie in the speed, the transfer rates on PCIe (the slot that graphic cards connect to) matter if you are going to have multiple cards. One card will be talking with the motherboard faster than the other. But again, if you are only going to buy 1 GPU and don't plan on any more, than ignore all of this.
As for memory size on a GPU. The use a GPU memory is very dependent on the resolution you play the game. More graphics memory is needed for High resolutions which is anything over 1080 (1080 is 1920x1080) like 2560 x 1440 or 2560 x 1600. When people play on high resolutions they usually have multiple monitors. You also have to take in account the GPU you'll be using. When it comes to high resolutions and games, you would want a very powerful GPU, we're talking like $350+ for a single card and most people have at least 2 of them.
Breaking it down. Here is the average frames for a few graphic cards playing BF4 beta. The link for the entire review is at the bottom.
So two things, firs the 7770 had 1GB of GRAM while the 630GT has only 512MB. The resolution here is 1280 x 720 on Low settings, which is significantly less than 1080. Just buy sheer power, the 7770 out performs the 630. GRAM had little to do with the performance here. You could also buy a Nvidia 650 GTX if you want instead of the 7770 HD.
As for the PCPartpicker incompatibility, it has to deal with a bios update to support the 6300. You may have to update the Bios to support the 6300 but that's also true for any FX x3xx series CPU. Again, you can change the motherboard back to the Asus one. I feel you like that one better.
http://www.tomshardware.com/reviews/battlefield-4-graphics-card-performance,3634-4.html