First I was going to suggest a hand grenade, but since you want to keep some it, I agree it could be pretty nice/retro if all those disparate elements went away. For it to work, at least for my tastes, I would:

1. Get rid of all the wood, and repair or replace the sheetrock that should be under it. Paint it an off-white or some very light color.

2. Replace the light fixtures and medicine cabinet, and get a larger medicine cabinet that will hide all the mounting holes in the tile.

3. Get a different vanity style and put the old sink in it.

As for the floor, you're right that it is likely to contain asbestos. The easiest, cheapest, and safest way to deal with it is to not remove it, but put a new seamless flooring over it if the existing tiles are well-adhered. If the existing tiles aren't well adhered, install 1/4" thick underlayment first so that you will have a sound surface for the new flooring to adhere to.

This is Daltile Desert Grey semi gloss, with a Daltile Chianti liner, and Daltile Desert Grey Speckle Keystones porcelain mosaic floor:

And Formica still makes this, which is their original boomerang, I think

And Wilsonart makes Ice Daisy, which is actually one of its 1970s patterns but it is similar to 1950s patterns

And if I went with a plainer countertop I might go all out with Cole and Son Fornasetti Peonie or Flamingos wallcovering: There is a better colorway of the Flamigos, I think.

I love that Formica boomerang and almost got it for my white laundry room, but the fabricator my builder uses only could do it in gloss and it would scratch too much in a laundry room (with baskets placed on it frequently). I could have special ordered matte but I would have had to purchase the entire huge piece. It's really a fun one.

Oh, man, with all the suggestions pal made I now want that bathroom.

I love pal's design boards! I also love that color porcelain with square pink tiles too. Happy you are trying to keep what works!

I would follow Pal's suggestions. He has a great sense of design

Wow, great ideas Pal!

Honestly, I am just reiterating what probably once was. :) My feeling is that if you are given a strong unusual element like maroon fixtures, unless you are going to replace them, you have to run with it. You can't "compensate for" or try and lessen the impact of something like this, it's too strong. Fortunately all three pieces of the suite are still present and Daltile makes a tile that works with it. No mosaic in maroon for the floor, though.

Love that you want to keep the red and are trying to get back to more authentic. Those flamingos make me happy!

Yes, go with Pal's suggestions! You can probably also remove the track from the tub. I have taken off a couple of those myself. They are usually screwed with two screws on each side, one top, one bottom. The bottom rail on the tub will likely be stuck in place with silicone caulk, and elbow grease will eventually get it off.

Does anyone have advice for shining up those maroon porcelain fixtures? What products should he use to try to bring back the shine?

Gold wallpaper I found under the Brown wood