Connections needs a “home page” because it needs to know where to send the search queries and category links for the Widget Pack. When using a template on a page, Connections will attempt to use the current page as the “home page” of that instance of Connections. I hope that makes sense.
To show templates on different pages, use the template shortcode option.
To limit what non-logged in users see, just use the visibility option for that item. Set it to “Private”. For example, lets say you have a phone number that is only to be displayed to logged in users, here’s what to do. Set the entry to be “Public” and when you add the phone number, set it to be private. Now, only logged in users will be able to see that number. NOTE: depending on the role that you’re logged in members are assigned, you might have to give role the capability to view private on the Connections : Roles settings admin page.
I hope that helps.