Probably for the entire wall.

Definitely the whole wall

Another nice option would be reclaimed wood, not necessarily rustic, but something with beautiful color or grain.

Stop at the edge of the stairs. Makes the "entrance to the room...rest is just staircase.

Stone looks best when it ends naturally- meaning it dies into a wall and look like an integral part of the home (wall to wall). It can look really cheesy and fake otherwise. We added limestone to our staircase wall. HUGE impact. Good luck!