I would probably use unmodified thin set to cover the gaps and give a good base for applying the tile.
Could you do something a little decorative and use a border tile around the repaired area, sort of like a picture frame? Then use a different tile to fill in the area. The border would make the change in depth harder to see.
For a backsplash, it's fine to tile directly to the drywall without anything behind it. For travertine, you probably want to avoid mastic. Mastic should NEVER be used on any natural stone because it is an adhesive and will discolor the stone (the adhesive has chemicals in it that will leach out over time). Use unmodified thinset. You'll also want to use a non-epoxy and unmodified grout (no latex additives) -- sanded or unsanded is fine. When the whole thing is done, you will want to seal it very well.
Travertine is VERY trendy right now, but you should be aware that it's not a renewable resource (it's mined limestone formations that accreted over thousands of years), the quality varies drastically, it is very porous, and it stains very easily. These things all make it close to my last choice for a kitchen backsplash. Even with a good sealer, it will still retain bits of food that hit it and will stain very easily if you splash something like tomato sauce on it. The sealer needs to be reapplied annually at least; preferably bi-annually. There are many porcelain products that look like travertine but are better suited to a kitchen environment.
As for how you'll mount it -- Decide on what kind of pattern you want. I have a soft spot in my heart for running bond with travertine. Mix a relatively small amount of thinset. Apply the thinset to the wall using a 3/8" notched trowel and rake it out. Stick the tile to the wall. Wait 24 hours for thinset to dry before grouting.
Pro tips:
- Use chalk lines on the wall to keep things level to your countertop.
- Cut all your tiles ahead of time if you can before you work on an area.
- Mix small batches of thinset, and only work a few tiles ahead. It sets up faster than you think.
- Keep a sponge and a bucket of water handy to wipe up any thinset that leaks between the tiles as you set them in.
- I really like Tavy Tile Spacers because they help you keep the tiles level to one another.
Best Answer
No you don't grout these period. I have installed similar 15-20 times. Impossible. Grout only effective if it covers, not if it is here and there.
So number 1 you can't grout this type of tile. They simply don't have big enough gaps. Even if you tried you would either miss coverage or by putting too much in you will distort the layout of some of the tiles.
Number 2... I really disagree with a lot of the comments. A kitchen is not a shower. Unless you are in a commercial kitchen, your personal home kitchen is not really a "wet zone". Who cares if something spills on it? You wipe it up. Oh no the magic tomato soup fairies are going to get underneath the tile and destroy it. It is absurd.
If anything it may in fact be easier to clean this tile. Tile does not stain or discolor from a spill easy if at all... what does stain and discolor? Yes grout.