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You have two ubiquitous handyman problems… the difficulty in finding a matching color for old grout and the deteriorating grout along the edges of a room.
Matching grout color is like matching paint! Seemingly slight color variations can produce glaring differences in repair work. One thing in your favor as compared to paint matching… the color of a powdered grout is the same whether it is right out of the box or completely set. However, when wet it might appear darker or lighter! Therefore you can very accurately judge the color without having to add water and pray! One other issue in matching grout is that the color changes over time due to simple age and reactions to cleaning products. Thus, you are really matching a discolored grout. Even if you knew the original color it might not be helpful in a repair situation!
The easiest way to match an odd grout is to mix two colors together and experiment! Adding so-called universal colorants is very iffy and I would avoid it. Granted, my method requires you to purchase twice the amount of grout, but may give you the most favorable results. In a shameless plug for "mom and pop" tile stores, I almost always do business with them because the folks there are the most helpful in these difficult situations. Many of these stores are run by people who were in the business and have insights you will not get from most "discounters".
The cracking grout around the room is a symptom of a common problem… the difficulty in sealing seams between dissimilar materials. The floor and wall expand at different rates and change during the year, causing lots of movement and stress at the joint where they meet… in your case, the perimeter grout line. There is no way to stop this movement which is why you will be making this repair over and over again. Though many moving joints can be successfully filled with a flexible adhesive caulk, this would probably be unsightly in your situation.
The better tile installers always remove the base molding before they install the tile (unless the customer objects, of course). The reinstalled molding does a good job of covering the edges of the tile and eliminates the need for grouting the perimeter of the room. Needless to say, this adds to the cost of the job due to the additional carpentry, painting, etc.
I have a suggestion that might address both problems at once. Have you considered installing a quarter round or stop-type molding on the baseboard along the floor to hide the gap? This would kill two birdies with one stone… you could use a flexible caulk of any color (or clear) to fill the gap if you want to and, as a bonus, neatly cover the gap!
In installing the molding, place strips of cardboard or aluminum flashing underneath it before nailing it on. This will provide a slight but unnoticeable gap between the floor and the molding making neat painting much easier!! Prepainting or staining is also a good idea if you can. Then you only have to do filling and touchups at the floor-line.
Tile grout is a product used to fill gaps between tile or natural stone products, such as marble. It is made primarily from Portland cement. That's right... the stuff that they make sidewalks and skyscrapers out of! Think of Portland cement as the "glue" that hold cement products together. Combined with other chemicals and mixed with water, a chemical reaction occurs which causes it to harden. The filler products that are mixed with the Portland cement determine the characteristics of the cement, such as how quickly it sets, how strong it is when fully hard, how waterproof it is and how much strength it has under various conditions.
With standard Portland cement tile grouts, there are only two types to be concerned with... sanded and unsanded. The added sand provides additional strength and durability to the otherwise fairly soft grout. Sanded grouts are used in work where the spacing between the tile or stone products is at least 1/8 inch. This minimum spacing is critical, because in thinner gaps the sand causes the grout to be unacceptably weak.
Unsanded grout is designed to be used in applications where the spacing between the products is very small. One typical use for unsanded grout is between ceramic tiles that have those little "nubs" to keep a uniform spacing between them.
Of course, there is a confounding issue. Most sanded grouts are labeled "floor grouts". This is a misleading label, however. If you install a floor tile that has less than 1/8 inch gap between them, the "floor grout" will fail for the reasons discussed above. In these circumstances, you must use an unsanded "wall grout".
Either type of grout is suitable for use with marble. However, the choice of color is more critical. Because marble can be porous, darkly colored grouts can stain it. Sticking to lightly colored grouts is the safest course!
There are well over a hundred different grout colors available from numerous manufacturers. If you have gone to one hardware or home store, you have not tapped into the variety that is available. I suggest you go to a quality tile store instead. Most of these stores stock one or two brands of grout in the most popular colors. However, the strength of these specialty stores is that they usually have samples of a wide range of grout colors that can be ordered for you. Just bring in a sample of the original grout for comparison purposes.
You do have another option and that is to mix two grout colors together. This is a standard way of obtaining an intermediate shade… good old trial and error!