Hi,
With the Starter edition, you will be limited to only custom address fields. However, with the Business edition, you will be able to add custom order fields which will be linked to the order and displayed on the checkout. You will also be able to add custom item fields which will be displayed on the product page and attach to each product in the cart (and then in the order).
It's possible to hide your custom address fields on some forms and not on others with a line of CSS. But the field needs to not be required is that case, otherwise, the system will refuse the form. If you give us the URL of your shop, we should be able to give your more precise directions concerning the CSS to use to hide your custom address field.