http://www.videocardbenchmark.net/gpu_list.php plug in video card in the search, write down the ranking and compare. If the difference between the 1 gig and 2 gig models, may as well get the 2 gig, although with that slower CPU you won't be running on high enough detail to matter.
Gaming is more dependent on the graphics card than the cpu.
If you are using integrated graphics, a discrete card will be a big boost for you.
I think your G1820 will be fine.
Think about changing it out only after you evaluate the performance with your new graphics card.
I would favor one of the GTX750 cards. They run quiet and require less power. Your cx430 can handle it easily. The performance of the GTX750 and R70260X will be similar.
As o 2gb vram vs 1gb, I don't know. I would spend the $10 extra.
The motherboard 1024mb memory will not be used since it is the discrete card that will have the vram. You will be able to disable the integrated graphics and/or reduce the motherboard vram to the minimum.
GTX650 and 7770 are older gen cards and are a bit slower.
Regardless of the card, you mostly are getting the performance you are paying for.