Power supply from PCI connector pin B25 or B5
Highly depends on your computer motherboard and power supply, as well as the specific FPGA (which you don't list) current demands.

That said, PCI is (according to both Linear and TI) has a hotswap, power management requirement. The PCI port is disabled unless a PCI card is plugged in.

You could simply take the power straight off the power supply. Most ATX standards provide generous 3.3v and 5v lines, so you can power it off 3.3v or use a regulator on the beefier 5v rail.