Are the walls taped and filled or skimmed?
A skimmed plasterboard wall will not hold as much weight as bare plasterboard and if tiling onto a painted surface the bond will only be as good as the paint to the wall
Also if you use the same tiles you will find that the flooring is cut thicker thicker than wall tiles so you will be overloading the walls.
Ideally you should completely remove the plasterboard and replace with new 12.5mm screwed to the studs or dabbed to the walls, in the shower area use Aqua panel and tank with WP1 kit from BAL
Grout is not the thing to use to fix tiles. You will be best off with a powdered adhesive that is white as grey will likely stain the stone, something like BAL CTF3 WHITE will do the job. The tiles should then be sealed with one of the Lithofin products and then grouted.
I assume the floor is 18-22mm chipboard. This is not strong enough to take natural stone, You have two options:
1, take up the existing chipboard and replace with 25mm WBP ply screwed at 200mm cts with noggins under all joints.
2, take up the existing floor tiles, overlay the chipboard with Aquapanel bedded in flexible tile adhesive and screwed at 200mm cts
You can then tile onto this with something like BAL rapidset Flexible(white) and grout with superflex or wide joint grout with AD1 admixture to make it flexible, not forgetting to seal the stone first.
Alternatively there are some very good ceramic/porcelain tiles that look like limestone /travertine around.
Jason