How can I clean and paint my basement floor?
Hard to tell what that white residue is from the description. It could just be old soaps or at worse efflorescence. Try using a mixture of 2/3 water, 1/3 bleach and some TSP.  mix a small batch and test a small area. Be sure to rinse it well and see it that gets rid of it.  If it doesn't do the trick, you may have to etch the exposed concrete with mureatic  acid and rinse well before repainting.

Assuming you finally get it cleaned up, let it dry well.  Use fans, dehumidifier or ventilation to speed the process. Then I'd really recommend a good bonding primer and overcoat with a good flooring enamel.  You can use regular concrete paint if it is not a high traffic area, but I like using a strong bonding coat first.