Yes, the GDDR5 will work fine.
When I said your socket (775) and ram (DDR2) were old, I meant they are old technology - your motherboard is from a several year old HP rig. Socket 775 is the Intel designator that tells you what type of processor will work. A 775 socket must have a 775 compatible cpu, just as a socket 1155 must have a compatible 1155 type cpu. AMD also does this with sockets such as AM3, FM1, etc.
The ram is also "categorized" in the same way. DDR2 is older tech and only works in m/b's that are built for DDR2 ram. Current generation ram is DDR3 and in most cases will not work on a DDR2 board (there a a very few exceptions). Both Intel and AMD sockets specify DDR2 or DDR3 for use. Both DDR2 and DDR3 are available in different speeds, but DDR3 is faster. With all that said, the GDDR ram type on the gpu does not matter to the m/b - it operates totally independent of the m/b.
Bottom line - your board will work with the 6670, but the limitations in speed of your ram and pci-e lane will not give you full capability with that card.