If your computer case is about 7" wide, then it is no doubt a full size (not slim-line) case. So you can use standard (not low-profile) gfx cards.
DDR3 SYSTEM memory is separate from GDDR5 VRAM. Yes, you can use the HD 7750 GDDR5 card in your system. In fact, it is preferable. The older GDDR3 VRAM is slow.
Since you have a full size case, there is one other card that you can consider. It too does not need an Aux 6 pin PCIe power cable from the PSU. That card is this one model (and only this one model) of the GTX 650: http://www.asus.com/Graphics_Cards/GTX650E2GD5/
As far as I know, Asus is the only manufacturer that makes a GTX 650 that doesn't need aux power. The others all need a 6 pin cable.
Here is a comparison of the HD...