Where do I store shipping and shipping material cost to be able to track that specific cost?
If you have the plus version or higher, you can create an inventory type item and stock the shipping materials that way. Then you can either include them on the invoice or use inventory adjust to lower the qty used. If you use them on the invoice, use a bundle item, that way they will not show and you put anything you want in the description block, ie
box, qty 1, price zero
cushioning, qty 5, price zero
description says, "thanks for shopping"
Be aware though, if you do you need to change the expense account to shipping expense, shipping to a customer is not COGS