id like to know, if i place a pci express 3.0 graphics card like the hd7870 in a pci express 2.0 slot, even running with a 22nm processor, that would mean the specs of the 7870 like pixel fillrate, texture fillrate or bandwidth will be cut by half?
No it does not in theory it would but no it would not. And this is why. Cards that are out now are not taking full advantage of the 2.0 slot bus. You wouldn't lose much if at all by having a pci-express 3.0 card in a 2.0 slot. So if you are debating whether or not it would be a good solution for you I'm here to tell you its a good solution.
id like to know, if i place a pci express 3.0 graphics card like the hd7870 in a pci express 2.0 slot, even running with a 22nm processor, that would mean the specs of the 7870 like pixel fillrate, texture fillrate or bandwidth will be cut by half?
Most GPUs use 16x slots for power delivery reasons. The actual amount of bandwidth necessary to feed mid range cards can adequately be provided by a 4x PCIe 2.0 link. High end cards can adequately be fed by 8x PCIe 2.0.
For most cards, there is no empirical difference between 8x PCIe 2.0 and 16x PCIe 2.0.
Similarly, there's no difference between 4x, 8x, and 16x PCIe 3.0
If your current mobo supports PCI-E 2.0, at this stage you don't gain much extra performance by going to a board that has PCI-E3.0. See this article where they test high end cards on both: