Bathroom floor

The beadboard is installed as is most of the chair rail capping it off. I primed the wood front and back. I then called in plumbers to run new supply and waste lines for the sink and toilet. I was expecting copper but who knew there was this new thing called pex, not me. I was a bit disappointed but there it is.
Here is the little toilet alcove. It is a really small space. A large person may not be able to wiggle his/her hips in there to sit on the toilet.

I put the underlayment down. I bought 2 sheets of CMX 1/2 inch plywood. I made a template of the floor area out of pink rosen paper, just a loose design. The main floor area is just under 4x8 feet so I trimmed the plywood to fit. I maneuvered the big sheet into the room and then could not get it down on the floor. I needed to slide the wood under the trim around the door but the heater vent and the shut off valves made it impossible. I ended up cutting the sheet of plywood in two. Of course the two cut ends matched perfectly and it was fine for the first layer. I bought two sheets of Luan for the second layer. I am unsure if I need to cut that into two. It should be easier to manipulate as it is only 1/4 inch thick.

The piece of vinyl I bought months ago is now laying flat on the basement floor. Again, I am going with the cheaper, temporary option. I would love to have hex tile but I don't think it was ever in this house. When tearing up the old floor, I uncovered a faux birck linoleum. This is also the first layer of flooring in the kitchen (under about 3 other layers of vinyl). Was the faux brick linoleum original? I am not sure. My house was built in 1935 and I have not seen evidence of it being around that early.