I have a question: I only want my admins to be able to see the actual Events listing page. If I set the dtregister menu item to "Special" then the public cannot reach the registration pages - is that normal? Does the menu item have to be set to public for the public to be able to see registration pages?
You are correct. That is standard way joomla menu items work. Whatever menu item is assigned to DT Register is what will be used throughout the registration process. So if you set it to Special, then public won't be able to access any registration.
Suggestion... make the menu item public but put it in a menu that is not actually shown on your site. Change the setting of the plugin to NOT show the "View All Events" button. Then if you just use Jevents and/or the EventLink plugin, the public will have no access to the list of events.
If you want admin to see the list of events, change the actual menu that you put the DT Reg menu item in... make that menu "Special" and publish it. Then that menu will only display for admins... but the actual menu item inside will not be limited.