What type of grout and mortar should I use for the compartment around my wood stove?
Yes, they have special epoxy mortar for heat sensitive locations. 

To me, mortar is what is installed on the back of each tile that fastens the tile to the wall. Grout is what is used to “fill” the joint between each tile. (BTW, this is exactly opposite of terms used for laying concrete block walls...grout is what fills the cells and mortar is what goes in the joint.)

Each “stove” manufacturer have requirements for “non-combustible” clearances between the stove and combustible materials. You’ll need to get the manufacturer’s installation instructions (possibly from the internet).

BTW, you asked about the wall, but you’ll need a non-combustible hearth too. Verify if your current wood floor can remain. 

Also, verify the size of your existing flue is acceptable, depending on fuel type, (i.e.: wood, gas, etc.)