So just to confirm, you only have a single unpublished site and it’s asking you to upgrade to a Lite Account plan before publishing it to the .io staging domain? Does the project have more than 2 static pages or 50 CMS items by chance?
In terms of adding hosting, you need to choose to add this before you can publish to any other domain. Webflow will guide you through the process—by showing you the following in the Publish dropdown within the Designer—but it’s not really “automatic”:
Clicking that will take you to the Project Settings > Hosting screen in the Dashboard (or you can go there manually) where you’ll choose the plan and then gain the ability to add a custom domain:
Regarding month-to-month hosting and transferring, once you initiate a transfer it will un-publish the project and disconnect any custom domains. That means that whomever receives the transfer will need to add a Site plan and custom domain after the transfer is completed. They would be able to access the site in the Designer, but not publish it to a custom domain (until they add a Site plan) or .io staging domain (unless it falls within the maximum pages/items in the free tier or they have a paid Account plan).