How long can I leave the water inside the tub? Do I need to dry it after each use?
The water should be removed when all the members of your family enjoyed their bath. Of course you can leave the water inside until the next day but unless you live in a very dry area, leaving water all the time of course can facilitate the insurgence of some mold stains.
Anyway, best practice is to empty the tub everiday (or at least twice a week if you have a recirculation system), quickly wipe the bathtub with a towel to remove the excess water. Then ensure there is a good natural ventilation (a bathroom with a vindow is strongly recommended) so the wood can adjust automatically its degree of humidity.