I'm going to request not to be charged shipping on an item
Why do you think you should not be charged?
There was a damaged package. The shipping fee should be refunded to the customer
It does sound like it was damaged during shipping and it would be bad to pay for a service that wasn't done well. It's smart to contact the company and let them know about the packing issue.
I don't like how they package their items.
Make sure to bring that to the attention of the shipping department. Some companies don't have guidelines in place for how an item should be boxed and it's helpful for you to let them know!