If I write a shipfic for A/B and C/D can I write a genfic for A & B & C & D focusing on their (canon) friendship, but have A/B and C/D feature in the background?

Yes, as long as the focus is on the A & B & C & D ship, that's fine.

And, yep, to the next - as long as you keep adding in or swapping round a character so it's a different combination, it counts as a new ship.

Good luck - I hope you have fun with the challenge!