I'm running electrical, Ethernet, and coax cable down the wall a fireplace is on. When standing in the room, it would go sheetrock, insulation, a gap, then the masonry block of the chimney. The cables would be installed along the stud against the insulation.

Does the cable need to be plenum rated? After doing some research, I am not sure if the plenum rated is only needed if the cable is in duct work. Which would be the case if this were a newer build and the cable was in the air space between the wall and the metal chimney pipe. As it is there cable will be outside the chimney air space.

Not a plenum so not required. Typical use is above ceiling tiles where return air is collected. It typically is Teflon jacket instead of pvc which is more heat resistant so you might consider using for durability but not for protection of off gassing during combustion.