There is a penalty for moving static colliders (colliders with no rigid bodies that they attach to), does that same penalty apply to character controllers? It uses a capsule shaped collider. Would it incur the same penalty as static colliders if used by itself?
Yes, there will be the same “penalty” which is not really a penalty but just wrong usage. Colliders can not detect any collisions. It’s the rigidbody that detects collisions. So moving a collider without rigidbody is not an option.
The CharacterController is a custom component that represents a capsule collider. The CharacterController actually uses some kind of raycasting to detect collisions that results of his own movement through the Move or SimpleMove methods. From the physics system’s perspective the character controller is just a static capsule that is updated and stored individually. That’s all handled by the CharacterController itself. You can not add more or different colliders to the character controller. Well you can but they would be just “dragged along” and do not contribute to any collision detection when the CC is moving.
In any case when you want to move a collider it has to be part of a rigidbody. This could be a normal rigidbody or a kinematic rigidbody. Of course a kinematic rigidbody does not perform any collision response on itself as no forces can actually act upon a kinematic rigidbody. However it can “wake up” colliding non kinematic rigidbodies which will act accordingly.
The CC is very limited and in that sense not part of the physics system. It uses the colliders in the scene for his own collision detection, but other than that it has nothing to do with the physics system. So the CC will always be an upright capsule. It can not be rotated or exchanged with a different collider.
To sum up: When you use a CC you only want to move the CC with it’s Move / MoveSimple methods. Of course for teleportation you can set the transform directly, but this won’t perform any collision detection. Never attach a collider to any moving object without having a rigidbody component on that object or on one of its parents. The CC is not a physics object and does not interact with them. It only stops its own movement when it detects a collision with another collider.