Laying plywood over floorboards for a vinyl floor
Underlayment for vinyl flooring isn't screwed down, as this would require that every screw be skimmed with patch material to create a flat surface. Ordinarily, narrow-crown (1/4") staples are used to prevent the need for skimming. Underlayment must be fastened on a 6-8" grid to prevent movement. 

No expansion gaps should be left. In fact, any gaps that aren't tight must be skimmed to fill and smooth them. Vinyl flooring does not span gaps without showing them through. 

Normally vinyl is installed, then base trim or base shoe is installed over the edges. It's caulked against the bathtub. Rarely have I sought to "waterproof" the perimeter of a room's floor. If you have that much water sloshing around on the floor, a lifestyle change is probably in order.