I would paint the vanity cabinet a deep navy. Then, paint the walls a greige color, leaving the wainscoting a bright white.

I would try for the lightest shade of blue. I hope you will be putting a glass door rather than a curtain so that you will be able to appreciate that tile. Side story: we just completed a master bath remodel and needed gray paint. All the popular ones from Sherwin Williams and Benjamin Moore clashed. We wound up taking a tile to Home Depot and it was scanned. They don’t really recommend this as a read on tile that has other colors through it as well as a hard surface, but we had nothing to lose. It worked!

Since you have experience painting over the floor tile, why don't you paint over the countertop, too. There are products made specifically for that specific purpose. If you want to keep the peach, I'd paint the entire room a color of white that works with the trim/cabinet/floors, and accent with navy blue rugs and towels.

Oh yes, a nice new light would be lovely. Even Home Depot and Lowes have nice choices for reasonable prices in lighting. Wayfair has a huge number of fixtures to choose from. I prefer ones with lights that shine downwards, but there are lots with lights facing upwards - that's really your choice as long as you get good light. Don't be tempted for any Edison style bulbs in the bathroom because in my experience they are often really yellow and that doesn't help anyone looking in the mirror trying to do makeup or shave etc if there is a yellow cast. I like this one which won't disappear against the wall. If you changed the handles on your vanity to match the dark or the bronze this might be nice but of course it depends on all your other choices. I thought this one was just a big different if that suits you. A little bit modern and industrial looking. Very classic and if I used this, I'd probably make the handles on the vanity match. you could also consider a new faucet in the same finish. For a floor pattern, you could likely do a hex stencil if it is small enough. I agree the existing pattern is going to make many stencil options not work for you. This particular one would allow you to paint small hexes throughout and then you could customize if you wish by adding a word like they've shown in the picture, or adding a border (that border could be the navy blue colour with the rest being white. Or here is another option where they've just used a plain hex in what looks a bit of a colour variation - that looks lovely as well. For your mirror, Home Depot carries kits so you can make a frame for your mirror. But someone posted on Houzz last week a place where you could order frames that are meant to dress up a mirror like yours and they were really nice looking. Worth doing a bit of research on that. Once you've decided on a flooring then that should inform your other decisions on wall paint colours. Would you consider a wallpaper or something like grasscloth on the walls in the bathroom. The grasscloth comes in some really gorgeous colours. Since it is a guest bath there won't be a lot use on a regular basis so I think you could easily choose wallpaper or grasscloth to bring in some colour. Look at these really gorgeous colours....if you went with a white floor either or these blues would be so lovely. But if you wanted to keep the floor etc as it is and used this taupe colour in the room along with your new white counter top and some bright white trim - that and a new light fixture etc might be all that is needed.