If I buy a PCI-E v2.0 graphics card (Nvidia Geforce GTX 275) will it work correctly and to the full potential with my PCI-E v1.0 motherboard (Asus P5B-Deluxe) ?
Yes. PCIe is backwards-compatible, so you can use 1.0 cards in 2.0 motherboards and 2.0 cards in 1.0 motherboards.
As to reaching the full potential of your graphics card, it's true that PCIe 1.0 is half the speed of PCIe 2.0. (So a PCIe 1.0 x16 interface is comparable to PCIe 2.0 x8.) But single graphics cards really don't saturate the bus anyway: if you dig around through older tech articles, you'll find a number of reviews of video cards in the x8 PCIe slots of early SLI/CrossFire motherboards. Invariably, the conclusion was that the slower bus didn't hurt performance noticeably at all.
You'll be just fine.
With a AGP 1x & 2x cards are only compatible in 1x & 2x slots. 4x & 8x only work in 4x & 8x slots.
You can not stick a 4x in a 2x slot.
There were AGP pro slots that had a workaround to allow it but it depended on whether the card could even drop down to 2x speed as most cards could not.
AGP is totally different to PCI-E in respects to backwards compatibility. The only time I have seen PCI-E graphics cards not working was because the BIOS of the motherboard did not allow 1x graphics cards.
As others have already said, technically, yes... but...
I seem to recall in the early days of AGP that the 1x, 2x, 4x, etc. versions were also supposedly all backwards compatible, but there were quite a few problems with 4x cards in 2x slots from memory (or maybe it was 2x cards in 1x slots).
Although it was designed to be backwards-compatible, the implementation was flawed and in practice it turned out there were some combinations that did not work as they should.