How do we manage products so they appear on our site without having any inventory?
You can do this by checking off the "Allow Backorder" box on the main page of any product.
By checking off the box and hitting "Save," the product will set all the items to "true," meaning they will disregard inventory and show on the site.
If you want to remove the product, you uncheck the box. It will reset the item to "false," and the product will only show if there is inventory. (This may take up to 15 minutes to be visible.)