I wouldn't paint the cabinet, unless you're doing it professionally. Change the wall color to something more neutral and bring the pop of color through artwork/accessories.

To me the wall, floor, and vanity are 3 different color looks, and look off together. The floor looks a little rosy, and doesn't look right with the teal. The vanity color is purplish and, possibly could have been a different color, to go with the floor, but looks like it will work with the floor with more coordination with the wall color. The wall is really out of place IMHO. Its too clean and bold with the dirty color tones of the floor and vanity. Get new wall color samples and try them out before you paint onto the wall. I always got large thick art poster boards and painted them instead of the walls. then I could prop them up and live with them for a day or 2.

White vanity, but probably easier to change the wall color...something in a blue gray or warm griege to relate to the vanity or floor.

I can see why you love the wall color. To me, the faded look of the floor is more out of place. A solid darker gray might fit better with the wall and vanity and tile as is.

A paint color with less intensity and more gray would work.

The floor is throwing me off. But I agree with some above statements, the wall color is the easiest fix. I do love that emerald green, perhaps you can bring it in through some accessories as mentioned above.

The vanity seems to have a different gray undertone than the floor, and also doesn't work with the wall color, so painting the vanity would be my vote to pull the room together. That is not likely an easy task, however, so consider whether you can pay to get it professionally done ($$) or have the time and ability to do it really well yourself. If you do go ahead with painting the vanity, why not go bold and paint it the same as the wall color?