1- No, the CharacterController doesn't use a Rigidbody (and attaching a rigidbody to it usually produce very weird results). You can use OnControllerColliderHit to push rigidbodies (see the example in the docs). To carry an object, you can child it to the character - but don't let it be in touch with the capsule collider, because it can disturb the character movement (it's a CharacterController bug).
2- The CharacterController has a built in capsule collider, and we can't change this;
3- The Move example doesn't allow control during the jump, but this can be changed if you store the vertical speed in a float variable and rearrange the code a little:

var speed : float = 6.0;
var jumpSpeed : float = 8.0;
var gravity : float = 20.0;

private var moveDirection : Vector3 = Vector3.zero;
private var vSpeed : float = 0; // keep vertical speed in a separate variable
private var controller: CharacterController; // controller reference

function Update() {
    if (!controller) controller = GetComponent(CharacterController);
    moveDirection = transform.right * Input.GetAxis("Vertical") * speed;
    if (controller.isGrounded) {
        vSpeed = 0; // a grounded character has zero vert speed unless...
        if (Input.GetButton ("Jump")) { // unless Jump is pressed!
            vSpeed = jumpSpeed; 
        }
    }
    // Apply gravity
    vSpeed -= gravity * Time.deltaTime;
    moveDirection.y = vSpeed; // include vertical speed
    // Move the controller
    controller.Move(moveDirection * Time.deltaTime);
}

This code only allows the character to jump when grounded; if you want it to jump even when in the air, move the jump code outside the isGrounded if