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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.
It depends on whether the grout product is latex-fortified, or the installer used a latex additive with a standard floor grout. The purpose of the additive is to add strength, stain resistance and color retention to the grout, making further sealing counterproductive.
If he didn't use an additive, a sealer will make cleaning a little easier and prevent some staining. It will have to be reapplied from time to time, though, and does slightly darken light colored grouts. Don't overlook this last point... a really white grout can turn very off-white after sealing, and the original whiteness will probably not return when the sealant dries.
A sealer can be applied after the grout is fully dry, usually within a week or two of installation. Try keep the sealer off the tiles, because dried sealer may show as an area of greater gloss on the tile. Any sealer that gets on the tiles must be wiped off while it is still wet.
I can only speculate that the sample displayed with the grout was incorrect or that the grout package was mismarked. Have you talked to the folks at Home Depot about this? It's worth a try! I have found the company to be very customer-oriented and perhaps they can give you some assistance.
Most colored grouts look different when wet. Interestingly enough, the darker the grout… the darker it looks when wet. A rule of thumb with grouts is that the color of the grout BEFORE you mix it (while it is still dry) will be the color after it is set. Look at the leftover grout (if you have any) and compare the color to the finished job… you will most likely find a close color match. If the color on the outside of the package is radically different from the color of the dry grout, you have even MORE ammo to get some sort of help from HD.
There are a couple of things you can do to change the color of the grout. The first is to apply a clear grout sealer. In my experience, clear sealers can tend to slightly darken grouts. Of course, this effect varies so I can't guarantee the effect on your grout. I would suggest mixing up a small batch of grout, let it set and then experiment with a sealer. The products are not prohibitively expensive.
Another option is to use a grout stain or colorant. These products, originally designed for sprucing up old grout, are available in a wide range of colors. After following the manufacturer's surface prep instructions, the product is either brushed on or applied with a special roller. Read the label, because there may be a waiting period before the product can be applied to new grout. These products are very durable and should look great for many years!