I would go with a very light gray, or even warm white.  If you are blond and use violet shampoo (it any tinted shampoo for that matter) beware of white grout because the violet can stain.  Recommend sealing your floor tile really really well once laid to keep it beautiful and stain free.

I would use a light gray grout. I have Calacatta marble tile which has gray, brown and white and the gray looks great and also seems to hide any potential discoloration as after 3 years of use, it still looks pristine.

I use appropriate cleaning materials for my bathroom which are safe for marble including products for cleaning my glass, mirror and toilet just in case someone isn't careful so there can be no accidents. I also use liquid soap in the shower and don't dye my hair although I always used liquid soap and never dyed my hair in the shower anyway. I personally would not use anything potentially staining in the shower like purple shampoo but you could probably do a test first and see if the particular formula is problematic.

Marble does have to be sealed periodically. It is sealed when it is installed and then periodically every year or so depending on usage.