" Also, cabinet guy says easier to use caulking to attach the tiles. Manufacturer is ok with this. Does this sound ok?

No. The TCNA manual does not recommend caulk as proper tile setting material.

Tile mortar is very easy to use once you get the hang of it. Mixing to the right consistency and using it up before it sets are important. I would apply over drywall assuming the tile is not too thick.

yes to the backer board behind the stove. It is needed for the heat from the stove. Since I did read elsewhere, that you are redoing a manufactured home, I am thinking maybe you need to use backer board under all the tile. I am pretty sure the manufacturer would never approve caulking tiles on. Read up on it and contact the manufacturer yourself. It sounds to me, like your cabinet guy is NOT a tile guy. You are buying fairly expensive tile, please hire a tile guy that knows how to install tile. Read this forum, you will see many tile disasters, because the wrong tile guys were hired. Dont let him touch your tiles, until you do the proper research on installation.