brown is another 'neutral' color that works with black and white scheme. Would be helpful if you can provide photos or links to the samples you have chosen.

The first thing I would do is switch your light fixture to track lighting. That way you can focus the light onto that wall much like what is done in an art gallery. I also think that may stop the reflection you are getting. I think putting cabinets on the long wall will really close in the space. I would make your red wall your gallery wall. I'd frame everything in white so they "pop" off the wall. The problem with the other wall being the gallery wall is that it's just to narrow. You can't really stand back far enough to see the art.

These chairs are easy to recover and the range of fabric available today gives you plenty of choice. I say go for it. If you don't feel up to it yourself get an upholsterer, but plenty of YouTube tutorials about.

Something to take into cosideration would be the amount of light you have coming into the bathroom. Obviously if you have considerable light and air the tile shade wouldn't be too much of an issue. On the other hand if the room is small and not very light filled, then the lighter shades would be preferable giving the illusion of making the room appear bigger and lighter.