Is there way to add extra fee for each order? More specifically I am searching a solution for adding packing cost for each order and this will be specified as an item on the invoice.
There are extensions in the Magento Marketplace, like https://marketplace.magento.com/amasty-module-extra-fee.html which will allow you to add an extra fee to the cart. The Amasty extension, for example, will allow you to make fees mandatory without an option for shoppers to remove the fee. Each fee you add shows up as its own line on the invoice, similar to how taxes and shipping show up as line items.