get a rigidbody rope move with my character

Try to parent it to the top of your ladder when you are in climb mode.

Add a new script to the rope telling it to move with the character. If you want a follow effect, attach t$$anonymous$$s script only to the top cube of the rope. If you want all of them to move at once, attach t$$anonymous$$s script to the rope's parent.

 public GameObject player; // get the player reference
 
 void Update () {
     transform.position = player.transform.position;
 }

I recently published an asset that could help you accomplish t$$anonymous$$s if you are still interested:

https://assetstore.unity.com/packages/slug/189679

Any feedback is appreciated!