Are the products created in the Console Cart module supposed to be related only to coaching? And what happens to the products if I leave the Console?

You can create in the Coaches Console's Shopping Cart module any kind of products that consist of courses, pages, forms, documents. They do not have to be specifically related to coaching. You can insert purchase buttons for these products on external websites (with the product code provided) for as long as you keep your Console account membership, and have your Shopping Cart module activated. If you close these, the purchase buttons for your Console products will no longer work.