Are you planning to keep the vanity when you remodel the bathroom?

As for quartz, it really depends on what you like. The Britannica is very nice, but I don't know if you would get the full effect on a smaller scale like the vanity.
We have Torrquay in the basement for the bar..it's a large area, but since there isn't much pattern, it would go well on small areas as well. It does have some tan/cream in it, so it's warmer in color. If you want to go towards a colder color such as gray, then you could pick another one such as the Swanbridge or Newport

your current countertops are Marble

Consider doing clarino from LG It has a soft marble veining in the warmer tones.

I really like Zodiac Valente Pearl

What you have now is marble - dark emperador.

I'm sure someone will use that top and sink. You can call the people from Habitat for Humanity to come and pick them up

It is unfortunate that they painted the cabinet. It looked really nice before.

If you want to keep the mosaic, you should go with a warmer color for the tile and the top. The gray one is beautiful, but it wouldn't work if you go with a warmer color for tile. If the mosaic has grays (it seems like that in the photo), then you could go with a white or grayish color for the tile and the top you like

You could ask the contractor who'll renovate the shower if they can remove the mosaic intact.

As for the top, if you like the gray top, which is beautiful, do that. It's a bit difficult to see how good the cabinets look now..they seemed to look better before, but painted, they seem to look a bit dated...sorry about the lack of better words..Not every style of cabinet looks good painted.

If you want less movement, try Frosty Carrina. London Gray quartz by Caesarstone is another subtle variety but more on the gray side. For marble look with more veining, check White Attica or Statuario Maximus from Caesarstone.