If you plan to install beadboard vertically, you will need to at least put blocking in the walls to nail it to. Wether you run vertically or horizontally, you need to put plastic on the walls first, then the beadboard. You NEED the vapor barrier to keep the beadboard from cupping and twisting.
Drywall behind the beadboard is fine. It makes a stout and quiet wall.
The drywall needs to be taped, but it doesn't have to be finished perfectly. Just get the joints filled flush and cover the nails or screws. The beadboard will lay against the wall better, smoother, solid as anything, easier to trim.
Welcome to the forum!