Your C2D won't bottle neck either card. If you are only gaming at 1280x1024 (less than 720p), either card will work well for you. If you want the best FPS, go for the HD 7750. Btw, there's no benefit going to 2GB of VRAM with a card of that class. Especially at that low resolution. It will barely use all of its 1GB.
But, if possible, look for a card with GDDR5, not GDDR3 VRAM. The memory bandwidth makes a big difference in frame rate.
Yes that is my case, and I have got a corsair 500W PSU the card you mentioned may not fit as its a tad over 7 inches I heard only under 7inches can fit properly, the Radeon 6670 is 6 something inches.. but the 7750 I mentioned the VTX3D GDDR5 one is more value for money but do not know this particular models size,,
Have you already measured to see? I know the similar Dell Optiplex has about 7 1/2" clearance to that shroud. But if you are sure the one I linked to won't fit due to length, then you will have to grab whatever low end single-bay short card fits. The HD 6670 is about the same performance as the on-die GPUs that come with the A10-XXXX series AMD APUs. The HD 7750 would be a step up from that, of course. What resolution is your monitor?
Yes, PCI Express is a backward compatible interface. No cards today can make much use out of PCIe 3.0 even if it were available. As to your situation... The lesser bandwidth of your MB's PCIe 1.0 probably won't make any difference with the slow DDR3 memory.
Previously I have measured it to be approximately 7 inches maybe a touch longer.. but I think it would be a risk if purchased the card that you linked maybe... my resolution is 1280x1024 I believe.. im not going for best image im going for fps... I dunno how bad my CPU would bottleneck a HD7750..