White lithium grease is fine. Marine grease but NO WD-40! That has cutting oil in it.
Rotating by hand is one thing, climbing up the shaft is another. You'll need to turn the gear in the opposite direction of what you do to just rotate it. It will automatically move up the shaft when you do this.
You may keep the boat dry in storage but you can't really do much about condensation when it's experiencing a temperature inversion.