Mini Calendar As Views Exposed Filter
You can achieve this with calendar module and Date API. The default mini calendar is redirecting to a new page, but you can override it in your template file. I would suggest to use ajax link api module to load your content in a div you would like. You can do something like the following

/**
 * Alters link url in calendar events block in order to filter events at /events
 *
 * @see template_preprocess_calendar_datebox()
 */
function &lt;themeName&gt;_preprocess_calendar_datebox(&amp;$vars) {
  $date = $vars['date'];
  $view = $vars['view'];
  $day_path = calendar_granularity_path($view, 'day');

 //$vars['url'] = 'exhibitions/calendar/' . $date;
  $vars['link'] = !empty($day_path) ? l_ajax($vars['day'], $vars['url'], '#block-system-main') : $vars['day'];
} 


l_ajax function is used to load contents via ajax. Let me know if you need any help.