So to fix the seat itself, should I look at getting it reupholstered with new foam padding? Or buy a new OEM or aftermarket seat?
I'll look into some weatherstripping for the time being, but I'd like to fix the problem at hand. The weatherstripping with the grooves should be able to slide onto the rear fairing pretty easily.

 

Either.

Even with a good seat, it won't seal tight enough to keep water out. If your concerned with things under there, you will need to cover or remove them. Some of the things are meant to survive a little water.