Creating a custom search form

Here’s an option.
Give a more specific name e.g. custom_search_form to your submit button:

(...)
//First a submit button
$output .= '<input type="submit" class="submit" name="custom_search_form" id="searchsubmit" value="submit" />'."\r\n";

Then you can add this to your functions.php file:

add_action( 'parse_request', 'my_custom_search_form' );
function my_custom_search_form($query) {
  //return if it not our custom search form
  if ( ! isset( $query->query_vars[ 'custom_search_form' ] ) )
    return $query;

  //handle form here
}

Also, don’t forget to set the query_var to true when registering your taxonomy:

'query_var'         => true,