I used to leave the lid off overnight for fear something would 'grow' if I covered the puck back up wet. Having pucks shrink and crack I began covering wet with no problems.

I strop on linen and leather before each shave and on leather only (20 round trips) after the shave. I never leave my straight razor in a humid environment. Always rinse, wipe with a tissue, strop, fold it into the scales, and put it in a drawer outside of the bathroom. I do not keep it in the original box. Personal preference.