Relay needs installing in the factory socket, passenger side footwell. Fuse is usually fitted, drivers side footwell.
You can use the existing socket located on the sonar sensor loom on the driver's side ( photos of all this is available on previous posts)
You can add two pins to the existing socket ( the side that attachs to the fog lamp, both the male and female sides were present on 3 alphards i have done, so just add two pins from summer other cheap connector from eBay)
I agree, the oem rear fog looks naff, but I'm sure it's actually the entire quarter panel that attaches to the rest bumper, so it's cost will be greater than valenti.
Edit. I was wrong about the quarter panel it seems, maybe I'm mixing it up with 10 series winter kit fog