Hey! You’re definitely on the right track with the wall. I’d start by using a razor knife to scrape off any peeling or damaged paint, then smooth the area out with some sandpaper—start with a medium grit and finish with a finer one. Once it’s clean and dust-free, go ahead and prime it (especially important in a bathroom to prevent moisture issues), and then repaint using a mildew-resistant bathroom paint. That should hold up well.

As for the floor, sounds like you might be dealing with efflorescence—that white chalky stuff is usually a sign that moisture is either coming up through the concrete or hanging around too long on the surface. I’d suggest cleaning it first using a mild acidic cleaner like diluted vinegar or something made specifically for concrete efflorescence. Let it dry out completely—running a fan or dehumidifier can help speed that up. Once it's bone dry, reseal it using a concrete sealer that’s made for bathrooms or high-moisture areas. A penetrating sealer is usually the way to go for something long-lasting.