You don't need to attach the plywood to the floor at all if you plan and cut the wood well.
I used a full sheet of 3/4" plywood that runs from the front passenger side, just behind the driving compartment. It was planned so that all the cuts in the plywood would be under furniture/cabinets and by screwing them down to the plywood, first, all seams would be hidden and secondly, both the weight of the built ins and the screws holding them to the floor would hold the plywood floor in place without a problem.
The second 1/2 sheet goes from the drivers side of the van and is the same length as the full sheet. The space at the rear is filled in from edge to edge and from the back doors to where the full length sheets end.
I did cardboard templates of the edges and marked the floor with tape/pen when I did the layout. A jigsaw was all that I needed to make the cuts.
As to the rubber mat, I hauled mine out because the underside insulating material was wet and nasty smelling. It also allowed me to clean the floor completely before putting in the flooring.
I used vinyl click imitation hardwood flooring rather than sheet vinyl. I find it much easier to install. It's also thicker and more impervious to damage. I found sheet flooring, whether at home or in the van, had a tendency to tear or gouge and nick.
