Couple of questions, do you know what type of walls are in use in the rest of you house. In construction from the 30's they applied a cement type of material on typically on a metal mesh for bathroom locations. I can't say I've dealt with 40's bathrooms. In my former mid 30's home the previous owners had removed old tiles and retiled over the base. For the most part it held good expect for an area where they had patched the opening for the recessed soap dish with a non water resistant material. I used a concrete patch to fill in the gap, and then used tile repair material available at most home stores. I had a tile guy come in for my 1/2 bath at that home to determine how to deal with some cracked and loose tiles and he mentioned that product is quite strong and it doubles as grout too. Just follow instructions.

This is indeed a rare, wall-hung dental cuspidor. They were first popular in the late 1919s when it was considered unsanitary to brush one's teeth where one washed the hands. They were first added by the wealthy. They gained popularity in the next twenty years and then as quickly were gone. Here are pics of two such dental cuspidors from a Victorian mansion that my sister is considering rehabbing. Simply gorgeous. These were not originally intended to hold water and most did not have facets, but rather spouts, and most did not have stoppers to allow water to be held.

I don't know of anyone who makes an "uncushioned" hex tile (one with a square edge) now. If you could find it, though, it would be installed the same as standard hex tile, and grouted with sanded grout. One thing you might think about-- if you can't find any that's uncushioned, there's still a way to get that same look. Use the cushioned hex, complete the installation (including grout), let it set for a couple of weeks, and then have a pro who specializes in stone refinishing come in and refinish your floor. What he'll do is actually use a progression of varying grits to grind the floor down, and by the time he's done, your floor should be perfectly flat, just like the one you have now.

Another vote for keeping the floor, but it needs work. Stains near the toilet make me think that it needs a really good cleaning. Or the grout needs replaced. If the grout is bad, get the floor evaluated for soundness. Is the underlayment rotted anywhere? If no, then you can get the grout replaced. If the grout is fine, it may be saved with the right kind of cleaning. Before you start, you might try getting a pet urine solution from the pet store (Simple Solution or Natures Miracle are the brands I see.) Soak the tile and grout with it near the toilet and then keep it damp with a towel that is damp with the solution. Leave it there a few days. My fruit stains completely disappeared in two days off a white shirt using this stuff. Gotta keep it damp, though. It is an enzyme solution that actually digests the urine. Later rinse and let it dry. If it works a little, but not enough, do it again. If it does not work, give up. Previous cleansers used there can make it so these do not work. If there are still stains, you may need to replace the floor grout. Make sure whoever does this knows what they are doing and won’t chip tiles! You will then have a gorgeous floor. The poster above may be right about the pedestal sink. The small vanity was added in the 1970s or later. Makes me wonder what there is no plumbing in the long vanity. If you are still there, we need a layout. That consists of a drawing with measurements of all walls, windows, doors. We can’t do much without that.