I did it several years ago, you just have to use the right adhesive.

Look for an adhesive that can handle heat. Many types of glue soften and release when they get very warm. Not sure what brand that would be, go to a real hardware store (not lowe's or home depot) and ask.

How are you going to tile around that millwork? Are you going to cut into the wood first, to allow a gap for the tile to slide under?

I don't see how this is going to look right without removing the mantel and the old surround first. It sounds like a klugy DIY job.

Your stone is miniscule. Seems like quite a bit of work to try to put stone over it.