No OSB! Home Depot and Lowes ought to have for sale plywood panels manufactured in Brazil or another South American source. However, do NOT use Meranti or Luaun plywood...junk. You want a multiply panel of either arctic birch or a close-grained equivalent from South America. Unfortunately, domestic plywood panels...even those rated for underlayment use...are not suitable.

We just had sheet vinyl installed (by a professional) and he went right over the subfloor. He never mentioned wonderboard or anything. It was really clean though, and then he had to prep the floor with some sort of compound first and let it dry for a few days before putting down the vinyl.

When using the pneumatic stapler you don't have to worry about length; it won't matter much if you hit the joists. 1" is a good length, even 1 1/4" is fine. You can use 3/4" if you want...all you're doing is laminating the underlayment to the subfloor. Also, don't worry about hitting a nail or screw below; most of the time the staple leg is deflected and bites adjacent to the obstruction. In those cases where you do hit a nail directly, the staple will just bunch on the surface and is easily removed. You should use the fastener schedule for the panel you buy. Most panels will have the schedule printed across the panel and at the edges. Just place a staple at every mark and you're good. Stagger the seams of the underlayment and the subfloor. I hope you've purchased an approved panel or intend to. Luan is not suitable, so if you've bought that stuff, take it back and get a birch or similar white wood panel made as underlayment. After your underlayment is down, only use a cementious patch/filler where there is a significant gap between panels. Simple light sanding to smooth panel edges to make them flush is all that is required. However, If the seam edges are pulled down enough to become a dip or trough in the surface, then use a filler to bring that up flush. Your success with this will be in your efforts to create a smooth and flat surface...that is your goal.

There are a few things to follow for plywood underlayment with glue-down sheet vinyl (which Linoleum, Congoleum, and other brands are all considered). The main priority is that it be APA rated, which means that it can take the glue (many use cheap lauan which is not suitable). Another priority is that it is fastened properly. The thickness is determined by the condition, levelness, and stability of your existing subfloor. The gap at the tub will be present with any flooring you choose, where the caulking must be properly installed and maintained. Your old house may have more movement and humidity changes that exacerbate that gap. APAwood.org is the go to for underlayment.