Should I install the vanity, or the split faced stone first?
I've installed that tile. It's not to hard to cut--even by hand as you can just cut through the mesh backing. Even easier with a tile saw. So I'd suggest not tiling more than you have to and live with the cutting. 

So I'd install the vanity first, then tile around it. 

That said, note that that tile a) can't be grouted and b) is very uneven and c) porous and rough. I'd suggest not using that as a backsplash at all as cleaning it will be a nightmare...and no matter how dainty we may think we are in the bathroom, the backsplash will get splattered when washing hands, cleaning makeup, brushing teeth, etc. I fear that you will regret using that type of tile as a backsplash.