The Ivy Bridge CPUs are all specified as supporting either one x16 or two x8 configurations. Does this mean that by installing the RAID card, the graphics card will only get to use eight lanes?
That depends on the motherboard and should be stated somewhere in the product description. You'll usually find it somewhere in the manual.
As a rule of thumb, if your motherboard has only two slots with an x16 form factor, both slots use the CPU's PCIe controller, i.e., they can be used as either a single x16 or two x8 slots.1
If the motherboard has more than two x16 slots, it's more complicated. Some motherboards split the lanes in three; in others, at least one will be controlled via the South Bridge and your graphics card will be able to use all 16 available lanes.
In the first case, the manual should state something like this:
The PCIEX8 slot shares bandwidth with the PCIEX16 slot. When the PCIEX8 slot is populated, the PCIEX16 slot will operate at up to x8 mode.
Beware that it is also possible that the remaining PCIe slots (i.e., those that are controlled via the South Bridge) share lanes. In this case, the manual should state something like this:
The PCIEX4 slot shares bandwidth with the PCIEX1_2/3 slots. The PCIEX1_2/3 slots will become unavailable when a PCIe x4 expansion card is installed.
Moreover, if I install a PCIe 3.0 card and a PCIe 2.0 card into the same machine, will the graphics card run in PCIe 3.0 mode, or will all lanes run in 2.0?
A PCI 3.0 card in a PCI 3.0 slot will run at PCIe 3.0 speeds.
I've seen benchmarks showing a slight FPS drop in some games between x8 and x16 - which I assume will only get worse as new games transfer more and more data back & forth.
This might be a minor issue for PCI 2.0 cards, but 8 PCI 3.0 lanes are rougly as fast as 16 PCI 2.0 lanes. The benchmark The Radeon HD 7970 Reprise: PCIe Bandwidth, Overclocking, & The State Of Anti-Aliasing doesn't show any noticeable difference.
1 This is not guaranteed to be true, but I have yet to find a counterexample.