In winter climates it’s extremely common to run humidifiers to add water to the air. Using a bathtub for the same purpose is pretty common and I’ve never heard of either causing mold in normal operations.
Yeah, given how dry our place gets in winter the added humidity is not a problem.
If I took baths every day then it might become a problem but once a week or so had had no negative effects.
Humidity is usually too low during cold months, so leaving the door open is fine and helps everyone feel warmer (humidity + a bit of extra heat).