Can I enter a product's price at checkout, like a shipping fee or a negotiated item?

Yes — make it an Open Price product. Turn on Open Price on the product edit page and leave the price empty; the cashier and product list show “Open Price” where the price would be. At checkout, tapping the product asks for the price first; enter it and tap Add to Cart to add it at that price.

• 0 is a valid price (free shipping, say); an empty field can't be added. Enter the price the way your store's tax mode prices everything else.
• A cart holds one line per open-price product: tapping it again opens that line so you can change the price.
• Turning on Open Price also turns on Exclude from promotions, so a fee isn't discounted and doesn't count toward spend thresholds. Turn it off again if a negotiated item should take part in promotions.
• Once sold, the line is taxed, earns points, and counts in Stats like any other, and a return refunds the price that was entered.
• A product with variants can't use an open price. Open-price products are never offered as gifts or paid add-ons, and never appear in the replacement list of an exchange.