When we were installing our wood stove, we were also in the beginning stages at that time, of a home renovation. There is a wall behind the stove that needed to be finished off.
First off, we had fire-retardant dry wall installed on the side of the wall facing the wood stove. Secondly, we wanted some sort of radiant surface, and like the look of brick (in our case) and so we had a professional Masonry person come out. We have him design freedom, only stipulating that he use brick.
What he came up with, was a backer board that is attached to the dry wall, and then (more specifically to your point) used half brick (thickness) with grout in between.
A couple of things here..........first off, using brick, seems to require grout, just because of the type of medium being used. It may be that slate doesn't need that, from a cosmetic point of view.
Secondly, I find in our particular application, that the wall behind the stove gets warm, but not HOT. I can put my hand on it during the hottest fire burn I can, without being burnt.
So this really boils down to what you like "visually" and what will hold up, long term.
I'm not a Masonry expert, so I can't say if slate without grout is less sturdy than slate WITH grout in between, but I CAN say that if your stove is one of the newer stoves out there, and if you are using Class A flue pipe (which allows you to be closer to the wall than a single wall flue pipe allows), you'll be closer to your wall than you realize ...............AND.................. you won't find ungodly heat build up behind the stove, causing worry over the integrity of the wall behind the stove.
Having said all that......with regard to using slate as a base ought to be fine. I wouldn't even worry about not using grout on that, as long as the slate is anchored in some way (I'm assuming there are multiple pieces, and not just one huge piece involved).
Then too, I was amazed at the fact that our Jack Russel could lay so long in front of our stove (relative to "heat") and so I went over and laid down at her level (basically putting my face on the floor), and was surprised how reasonably cool it was, at floor level! I'm not convinced the floor material is going to get excessive heat build up (heat rises, obviously) so it's really only about floor integrity, and resistance to burning.
I don't think I'd want my stove ("static" as it may be) to shift on me because of loose or non-anchored (grout or otherwise) base materials.
-Soupy1957