For cleaning, I'd start with a dust mask, head lamp, and big plastic brush, followed by a shop vac.
Then, squirt bottle of water and shop vac to pick up the water on the floor, followed by de-humidifier.
If you are constantly getting stuff off the walls, you might be causing a reaction to something with your cleaning agent so "getting stuff off" isn't an indicator of being clean. You are just causing a reaction and then getting the reactants off the wall.
Basements need a constant barrage of "clean a few spots a week" to stay nice.
Once you get the water issues fixed on the outside, consider a nice bright coat of paint. Painting rafters also helps brighten things up a lot, keeps light bouncing around rather than just getting sucked up.
Only after the water is stopped would I use bleach, or on existing painted walls.
Use active ventilation, either run your AC / Forced Air on "Fan On" or use portable fans.