If the door is well painted I doubt if you will have a problem. A good coat of paint on wood work and wall surfaces is important if you have a lot of moisture in the room. If you can do some alterations to the space I would suggest you install an exhaust fan in the ceiling that vents moisture to the exterior, i.e. through the roof or side wall. Companies like Nu-tone and Broan make fans that have a moisture sensor that will kick on the fan while using the shower if you forget to turn it on. And these fans can be much more quiet than the typical bathroom fan. Some hundreds of dollars for the purchase and installation of this would be well worth it. I have seen bathroom ceilings covered with mold because the room has neither a neither a fan or window so you are right to be concerned.

HI -- I have the very same bay windows . I put Hunter Douglas wood blinds fitted to the windows and have hanging drape side panels on the side walls on each side of the bay . So nothing hanging in the middle . I have the panels ( 95 inch length ) hung from the full wall height (8 ft.) on a rod that goes across the width of the whole wall. Or you could just have the rod the with of each side wall ,having a finial on the interior ends . So that would be ,two small rods with 2 wall cup brackets and two finial ends . I am not a fan of the shutters because of the cost but mostly because they block too much of the light and view . Have the window blinds and the bay interior the same color .

'Supreme" chairs come in dazzling colors.I flipped for them when in Kerala but bought them while at Vijayawada. They are available anywhere and are water resistant, strong and too cute to look at. I am yet to find a table to pair them with but will do so soon. red,lime green,black,white,brown,blue,yellow...my god.The colors and their vibrancy is awesome.Many more colors as listed are available.

You would need a base steel grid in case of a climber. Plants that require least watering would be bougainvillea, blue morning glory, star jasmine, bleeding heart. If you are not keen on a live wall perhaps you can install artificial grass/turf on the wall and complete with some wall masks.