Does the bathroom have an exhaust fan? Does it get used during and after every shower?

Peeling paint could be due to bad batch of paint, poor wall prep before painting and/or a humidity problem.

I'd guess you put latex paint over old oil-based paint without appropriate wall prep (sanding) or an intermediate primer like BIN that adheres well to both. Humidity might well also contribute. I'd probably scrape, peel, and sand to get off any of that paint layer that is not thoroughly adhered, prime with a shellac-based primer(BIN), then the appropriate base coat for underneath wallpaper.

I don't think there's a great answer for this. Left undisturbed the wallpaper would probably have stayed in place fine, the danger is that the uneven surface will telegraph through the paper. Depending on the weight and pattern in the paper that may not matter. You can also apply a liner paper before your decorative paper that will hide the unevenness.

Invest in a good bathroom fan (Panasonic makes nice quiet ones) and wire it with an occupancy sensor and timer so that no one has to remember to turn it on.

Do yourself a favor and skip the peel and stick stuff; it's actually easier to get a good job with wallpaper paste --you can move the paper around before it sticks in place to get the alignment perfect-- and it won't start peeling off around the edges in a couple of years.

rwiegand said it all and said it well. Follow those instructions to the letter and before painting anything else ALWAYS determine EXACTLY what you will be covering up.

" I'd guess you put latex paint over old oil-based paint without appropriate wall prep "

Yep, classic peeling pictures from this. Comes off in strips or pieces, not just peeling.

When I look at your picture, I see paint that is not bonded to the previous paint on the wall. It appears that maybe at one time, your bathroom, was painted with oil based paint, and then repainted with latex, without the proper preparation of cleaning, and applying a coat of a sealing bonding type primer before the latex. The only fix, is to peel and sand it all down, so its smooth again, and this time, use the sealing bonding primer, and two coats of good quality bathroom paint. If you dont have a fan in there, its time to put one in, and make sure it vents to the exterior of your house.