You will need a wet tile cutter if you want it look seemless and perfected, nothing else makes such a clean cut.
Yes, you're right with the sealer, to stop the grout soaking in the tile and making a mess of it, you will seal the tile before you grout, some do seal before they even lay them but you need to go over the the cut edges again if you do it that way.
Adhesive and grout is cement based and can usually take heat as far as I am aware, just double check this on the bag. Colour is usually your wifes choice

though.