Sumod- The medicine cabinets are simple wood frames with a cabinet-grade plywood back that just sit in the stud cavity. Because of the dimensions, the front of the frame projects about 1"+ beyond the finished wall. The medicine cabinet doors are also plywood with veneered edges and they are attached to the frame with standard cabinet hinges (Euro hinges). The mirrors are just 1/4" plate glass ordered to size from a glass shop and held in place with mirror mastic and a metal strip at the bottom that you don't really see. The guy who did the cabinets just made the medicine cabinets as part of the job. I don't know about the door brand...we live in a large metro area that has lots of door shops and the doors are just solid core maple doors with the center cut out and a maple panel held in with molding. They're not stained; they simply have a couple coats of clear spray lacquer.

You should absolutely seal ALL natural stone you use anywhere there is moisture. The watermarks you have right now should dry up on their own, if they do not, they can easily be removed with baking soda mixture. Once the color has been restored, use any sealant that is suggested to seal marble (HomeDepot has many options and someone will be able to assist you). Check to make sure the tile is still sealed every couple of years and reapply as needed.

I totally understand your concerns about maintaining congruity with the era of the house. My personal feeling is the the mix of old and new is what makes design interesting. I'm not a fan of walking into say an all modern room in a vintage home. It's jarring. But intentionally integrating a modern element into a more traditional space can be the spark that "makes" it. So, for your bathroom, the larger hex can be a "riff" on the classic tile - that updates it while playing homage to the house's era. At the same time, the use of a solid blue on the floor is already a more modern use of the material. So you could go with a smaller blue hex rather than the larger. I would choose your grout very carefully. I personally don't like a ton of contrast between the tile color and the grout. It just looks really busy to my eye. I also am imminently practical and would never put white grout on a floor. Something in the medium gray range would probably work best but I would have your tile installer make you up a couple sample boards once you get to that point. Smaller hex in a solid color: Grout similar to tile color:

I wouldn't put wood ceramic tiles in kitchen since you are having wood look laminate elsewhere. I would opt for a tile that you like. Are there inspiration photos that you are drawn to. I am not sure what regulations are for your condo in terms of sound abatement. In my experience, there are no soundproofing regulations for bathrooms and kitchens in condos because these rooms were not intended to have carpets or rugs and were generally originally built with some kind of hard flooring. There is probably more noise carried from showers and toilets than people walking in the bathrooms.