I would think that either 14 or16 would probably be adequate. I think that it what most of the long lamp extension cords that I've seen are 14 and the shorter ones are 16, but they are usually sized for a 1K lamp.
I think that would work fine to make the ballast/relay connection. My preference would be to terminate all of the wiring whose origin/destination is outside of the enclosure on terminal blocks and then on to the various components. It simplifies testing and parts replacement later. Are you planning on mounting the ballasts inside the enclosure? If not you will want some kind of strain relief for the cords where they enter the can.
I think that would work fine to make the ballast/relay connection. My preference would be to terminate all of the wiring whose origin/destination is outside of the enclosure on terminal blocks and then on to the various components. It simplifies testing and parts replacement later. Are you planning on mounting the ballasts inside the enclosure? If not you will want some kind of strain relief for the cords where they enter the can.