Can I manually add a new page to the menu structure?
Sure can! There is a simple menu.php file that contains the unordered list that you can add any additional links to. For some more functionality when modifying the menu, you can use CSS Menu Writer.
If you select News and Events on the left side as highlighted and then select "view this page" under where it says "choose the content area you wish to edit below"
Actually this is as designed, but i can understand your confusion. Let me try and explain the logic behind the templating feature of the CMS a little more...
The Template page (the page you specified in SS as a content management template) is not intended to be the public page of your site as much as just a template page to define where content goes for the pages based on that template.
For example, if you had a news website, you could have a template named "articles". The template defines the layout of all the article pages that a user would add to their site from within the CMS. This also defines the URL for the news section of your site (articles.php).
Here's where the template feature can be a little confusing, and where you are a little confused....
If you haven't yet created a new page based off the articles template, you can navigate to that URL and you will see the content of articles.php. Once you have added a page based off that template things change a little. Instead of showing the content page when browsing to articles.php, you are taken straight to the most recent page added (in the case of a news website...the most recent article). One way you could use this is by having a recent articles link on your site, and it will always jump straight to the most recently posted article.
Hope this clarifies things a little further for you.