How to split up the renderd page in a tpl file?
$page is the rendered content from page.tpl.php - you can't alter it in the HTML template without resorting to nasty string manipulation.

Fortunately your desired output lends itself nicely to being handled in the page template anyway. You won't automatically have the same template name suggestions available, although a theme/module might already make them available in your installation. If not, you can add it yourself:

function MYTHEME_preprocess_page(&amp;$vars) {
  // This condition may need to change depending on your exact needs
  if (arg(0) == 'events' &amp;&amp; !arg(1)) {
    $vars['theme_hook_suggestions'][] = 'page__events';
  }
}


Then in page--events.tpl.php you'll have access to the individual elements of the $page array, and you can render them as needed.