Drupal 7 views exposed filter: Removing -Any- and sorting items
You should implement hook_form_alter on your exposed filters form (if you have devel enabled then use dpm to find the form's name).
Using the form alter sort (or just revert)  the date's array and remove the '-any-'. 
By the way to remove the '-any-' you may also use CSS which will allow you to quickly bring it back in the future if you'll want to.

I edited a similar code I used in the past according to your need, so you will need something like this:

function mymodule_form_alter(&amp;$form, &amp;$form_state, $form_id) {

  switch ($form_id) {
    case 'views_exposed_form':
      if( !empty($form['#id']) &amp;&amp; $form['#id'] == 'views-exposed-form-myview') {
        // This is the myview view

      // Remove the start date label
      if( !empty($form['field_sw_date_value']['min']['#date_label_position']) )
        $form['field_date_value']['min']['#date_label_position'] = 'none';

      // Remove the end date label
      if( !empty($form['field_sw_date_value']['max']['#date_label_position']) )
        $form['field_date_value']['max']['#date_label_position'] = 'none';
      break;
    }
}