If a non-logged user creates a “Save for later” list, will the list be lost as soon as the browser is closed?

No, the plugin stores all data from the list in cookies for 30 days for non-logged users, while logged users can recover them any time they need them because data is stored in the database.

Can the shortcode only be used in the “Save for later” page generated by the plugin?

No, the shortcode can be added to any page of your site. You can read more about how to use it here.