It took me a day or so to figure out your question(s). I think this is a difference between the behavior you expect versus the actual Plans behavior. Let's start with your first question:
Show/hide tabs is an always/never deal. There is no "if I enter the URL of tab 2 in my browser the tabs will appear" if you have them disabled. Therefore, if you want to get to tab 2 you will need to enter the exact URL. If you have tabs disabled you'll need a "cheat sheet" to get to the tab you want":
Copy/paste that into a new HTML file, save it on your computer and use it to link directly to your desired "tab".
Second question about Login/Logout:
This does NOT log you into an "admin panel". All it does is save your calendar password for the current session so that you don't have to keep entering it on each add/edit/delete. On clicking the "Login" link you should see a popup that asks for your password. If you don't get that you've deleted/commented something that's affecting that popup/prompt. The other possibility (a very outside-the-box possibility) is that displaying your calendar in an iframe is somehow affecting the javascript's function. In other words, it can't figure out how to display that popup within the iframe. Try opening your calendar in its own page/tab/window outside the iframe and see what happens.