From what I remember, marble is a form of limestone and you cannot use any strong cleaners on it, such as bleach, ammonia or other store-sold cleaners. You are doing the right thing damp-moping with water.(You could add a squirt of milod dish detergent). Maybe someone else can tell you how to clean grout.

Sounds like a mold. After you clean the grout, seal it.

My daughter contacted her marble tile processor and bought a cleaner they recommended. Her shower was a year old and whatever she was using was not removing the soap residue.

I see a couple of brands of marble cleaners / seals at Home Depot. Are these reliable or are there special brands to consider?
I use a shark mop to clean the marble floors regularly, but the seal has worn out and they are now porous/opaque and absorbing a lot more dirt.

I read about the bona stuff, not sure that is specifially great for marble, seems to be very highly rated for wood floors. Not my case. Any advice?

Steam. Check out the article.

Here is a link that might be useful: Steam Cleaning Marble