I would introduce gold hardware....go with a deep blue or teal painted vanity with gold hardware. Yes match the white in the tile for upper wall...remove cabinet over toilet....replace with a great piece of fun wall art.

old dresser with white vessel sink?

Yes, try to match paint with the tile. I would get a new vanity with sink and put a frame around the mirror!

If this is a powder room used by guests, why not consider a nice wallpaper? The room needs some color. If the tile is in good condition, you can echo the gold flecks of the tile in gold color hardware. You can use the colors in the wallpaper for hand towels. A remnant of Quartz, granite (or another solid material) counter would not be too expensive. Most granite yards have remnant pieces at decent prices and will cut them in the size needed. Check out Houzz photos for "powder rooms" for other ideas. If you gave a particular style in mind, add that term while searching Houzz.