As CoCo and Geof already answered, your 800 should be fine. You connect with the supplied cable(s). It's a circular threaded type connector which all mid to high level flash systems use. The threaded connection reduces/eliminates accidentally unplugging the unit.
Yes, it comes with a cable to go from the 3.5mm jack on the PW-III to the PC sync socket on your SB-800. It also includes a double ended 3.5mm cord for connecting to things like alien bees strobes or anything else that uses the standard 3.5mm jack.
I believe so, the sb-800 has a PC sync port which the Plus iii comes with all of the cables for that. If the port does not work for some reason you can always get a cheap hot shoe to PC to plug into. At any rate, I think it should work alone with the 800 though.