Every Piece of movement is clean and appealingly playable. Really really nice piece of work. As a Unity Game Developer I will suggest you two things.
1- When you enter cavern you can disable rotation of player with respect to camera, its feels weird when you rotate the camera, your rotation changed and player might don't know where the player is facing now. And the movement keys are now randomly face the player.
2- You can also disable the wall detection from player front side for wall running. Its becomes annoying when you are jumping small stairs like platform. bcz it sticks to the wall if you are close to it. Or you can define only some wall to be wall runnable.
These are just from a dumb guy so don't lose you creativity on my suggestion. Maybe you have better approach than this in your mind.
Just Consider this as suggestion.
Love the game and Good Luck.