How to paint 'Chimney' on the rooftop successfully?

Ok, I'll admit that I'm a relative newbie, but I've done a fair amount of research and am still stumped. My wife and I just purchased our first home, a 1960s rambler with what I am assuming is original wallpaper in the bathroom. Other than the wallpaper from the top third of the wall, the rest of the bathroom is tile. So, the first thing that wallpaper removal sites tell you to do it just grab and pull and pray. So, I did, and the wallpaper was nice and vinyl and pulled right off. But, I have NO idea what is underneath it. It's not drywall. I am guessing that based on the ceramic tile in the rest of the bathroom, that it is backerboard. Is that right? If so, how would I go about painting it? Do I need to use drywall mud to skim coat it and then sand? Or am I going about this the wrong way? Thanks for any suggestions.... Here are a couple of pics. The bottom pic is the whole bathroom, and you can see where I started to peel. The top pic is a picture of what's underneath the wallpaper as I pull up.

Just a little backstory- the doors to the house in question are OLD, the house was built in 1922 and I am assuming all the doors are original. They were stained a very dark cherry/ebony color and then varnished subsequently many times. A few years back they were "stained" - and by stained i mean that the stain (also a very dark color) was put on the door and left on to dry over a few weeks erroneously. Fast forward a few years later and the doors were wanting to be painted white- so they were heavily stripped of stain/varnish layers with Citri-strip gel, cleaned, and then painted with Behr Semi Gloss paint- a few coatings of it. The doors look great at first but after a few weeks pink wood grain patterns (bleeding) come through. They have been painted a few more times in the same white color and again the pink bleeding in wood grain pattern comes through. At this point it would be absurd to strip these down and start over. Is there anything that can be painted on top of the existing paint layer to prevent the bleeding and THEN repaint in the desired color? Any suggestions would be greatly appreciated as this is endlessly frustrating. Thank you