The shower looks large enough that it may not have a door- the panel might be fixed just to allow natural light in. I can't tell if there are hinges or not from this angle. The photo might be taken from the door opened- you can see a curb at the bottom.

That looks like a solid slab of stone that's been cut out..possibly limestone or slate?

Many people enjoy the feeling of pebble stone underfoot as it feels organic, natural and like you are outdoors, while others don't care for unsmooth surfaces underfoot. This is most likely a personal preference.

There are pebble floors that have a smooth even texture to them. I have used them for clients that have sensitive feet and do not like the uneven feeling of some of the pebble floors.

What about cleaning? We have hard water and poor ventilation in our bathroom and I worry about mildew and water deposits btw the stones?

I'ts no worse to care for than any other tiled floor

Using Epoxy grout helps a lot with mold and mildew. However, most installers will usually try to talk you out of it because of additional work required.

shouldn't be a big problem if it is sealed properly.