So I have painted a lot and I mean a lot of that kind of beadboard. It is definitely time consuming but doable. Here is how I felt worked the best for me. I used a combination of a brush and a small about 2 inch foam roller (to get the smooth look on the bead board). First I feel it is easier to paint the beadboard first on the wall and then put the baseboard and trim on and paint that and tape the bead board after it is dry. I use a combination of a brush and roller. I work in sections quickly (from left to right) and move along the wall. First I use the brush with paint and paint from top to bottom in the grooves. This will get the paint in the grooves of the beadboard that the roller won't. I usually use qet a bit of paint in the brush. Then I go behind the brush and use the foam roller going in an up and down motion (V motion) from top to bottom and move a long the wall. The roller will provide a nice smooth finish without leaving brush marks. Then after I have done one complete coat using the brush following quickly with the roller, then the next section brush following quickly with roller, until I am completely done. Let the whole project dry. Then I go back with a 2nd coat using only the roller. Let that dry. Then I attach the baseboard and trim. Caulk the cracks and nail holes and then paint once more. (I paint the trim and baseboard on the floor separately twice so it only needs one coat when I put it up) I usually will paint the baseboard one more time to cover all the nail holes (but that's just touch up). Good luck!! It will look great!

As others have said it's hard to say without seeing pictures, but, one thing you said makes me wonder if the beadboard look will work. You mention that this is a corner. Does this mean you would only put beadboard on two walls? IMHO it needs to tie in to the surrounding areas. I don't think I've ever seen it used on just some of the walls in a space. I actually looked for this when I was remodeling my open plan basement. I really wanted to use beadboard but couldn't find a logical way to use it on some walls and not others.

Using oil based primer means putting up with extra drying time and more odor. I am not a pro painter, but do painting as part of my business and learned from pros. What I do for interior wood is to use Zinsser BIN or Bullseye 123 primer---you can use water based BIN for your application. That will make much faster drying time and much less odor. Use a matte latex. But, an eggshell or semigloss will be easier to clean Tint the primer with the paint color, that helps coverage and color intensity. Red is the hardest color to get even color/coverage, IMHO. However, it will allow you to vary the coverage and get a distressed look---you will see this happen as the first coat of paint is applied and dries.

I wonder how wainscotting will look with the odd angles. It might accentuate how non-colonial the powder room dimensions are. I am no decorator, however and may be wrong. I do think that once you are done, you will wish you had replaced the toilet, too. It is more rounded, and the sink's sharp points and shiny finish may clash with it. I wonder if wallpapering the wall behind the toilet and the short wall behind the door , while painting the other two walls in semi-gloss (in case of splashes) might be a good idea? Or better, wallpapering the lower part of all of the walls with a non-vertical (like floral or all-over patterned) wallpaper might be better. I am thinking that it is the vertical stripes of the wainscotting that will draw attention to the room's shape, while a patterned wallpaper with a chair rail will not, so much.