No, by default the picklist in Pickware for Shopify does not show any weight, neither the item weight per line item nor a total weight of the order. However, you can add your own weight column with the document editor, without entering the weight again for every order.
What is on the picklist
For each line item, the picklist shows the columns Bin location, Product number, Product name, Quantity, and Done. Data on weight, dimensions, or packaging is not included.
Add weight to the picklist
Note: The document editor is included in the Professional and Enterprise plans as well as in the trial period. You can request a free trial via the Design the document button.
You add a weight column in the document editor. To do so, open the picklist in Pickware Admin under Settings → Shop → Documents and click Design the document.
The easiest way is to describe the change with Pickware AI, for example: "Add a Weight column to the table with the weight of the product." You can find out how this works in Customizing document templates with Pickware AI. If you are familiar with Twig, you can also customize the table yourself in Expert mode, see How can I customize my order documents with the document editor?
Where the weight comes from
The picklist takes the weight from the product data in Pickware. You maintain it in Shopify in the product properties under Shipping → Product weight, and the product import transfers it to Pickware by default. Where you maintain which product data is explained in Where can I maintain which product information?