Would you want the content on those paginated pages not indexed? It is possible to tell bots not to index pages with search and filter query strings on them. You could do this using the functions.php file of your child theme, maybe something like this:

add_action('wp_head', 'noRobots');
function noRobots() {
  if(strpos($_SERVER['REQUEST_URI'], '?_sf_s') !== false) {
    echo "\t<meta name='robots' content='noindex, nofollow' />\r\n";
  }
}

To the first question, I do not know. You would need to try it and see what is output on the page and see if you do have both sets. If they ARE both there, I suspect that the page will not be indexed. But you may need to ask them that question. They may have a better way of doing it.

I am not sure what can be done about the situation where the search has no results, but you say post show anyway? But, if the noindex thing works, is this an issue? Or are the terms still in the dropdowns, in which case …

In the General settings tab of the form:

Set Auto Count (both settings) to ON

And then in the Form UI:

In each field set Hide Empty to ON

That issue is simply the way that whatever code outputs the posts handles a ‘No Results Found’ scenario. As you are using our Shortcode method, and thus the infinite scroll version of results.php, this customized version handles ‘No Results’ better I think:

<?php
/**
 * Search & Filter Pro 
 *
 * Sample Results Template
 * 
 * @package   Search_Filter
 * @author    Ross Morsali
 * @link      http://www.designsandcode.com/
 * @copyright 2015 Designs & Code
 * 
 * Note: these templates are not full page templates, rather 
 * just an encaspulation of the your results loop which should
 * be inserted in to other pages by using a shortcode - think 
 * of it as a template part
 * 
 * This template is an absolute base example showing you what
 * you can do, for more customisation see the WordPress docs 
 * and using template tags - 
 * 
 * http://codex.wordpress.org/Template_Tags
 *
 */

if ( $query->have_posts() ) {
 ?>
 
 Found <?php echo $query->found_posts; ?> Results<br />
 <div class='search-filter-results-list'>
 <?php
  while ($query->have_posts())
  {
   $query->the_post();
   
   ?>
   <div class='search-filter-result-item'>
    <h2><a href="<?php the_permalink(); ?>"><?php the_title(); ?></a></h2>
    
    <p><br /><?php the_excerpt(); ?></p>
    <?php 
     if ( has_post_thumbnail() ) {
      echo '<p>';
      the_post_thumbnail("small");
      echo '</p>';
     }
    ?>
    <p><?php the_category(); ?></p>
    <p><?php the_tags(); ?></p>
    <p><small><?php the_date(); ?></small></p>
    
    <hr />
   </div>
   
   <?php
  }
 ?>
 </div>
<?php
} else {
 
 //figure out which type of "no results" message to show
 $message = "noresults"; 
 if(isset($query->query['paged'])) {  
  if($query->query['paged']>1){
   $message = "endofresults";
  }
 }
 
    if($message=="noresults") {
    ?>
 <div class='search-filter-results-list' data-search-filter-action='infinite-scroll-end'>
  <span>No Results Found</span>
 </div>
 <?php
    } else {
 ?>
 <div class='search-filter-results-list' data-search-filter-action='infinite-scroll-end'>
  <span>End of Results</span>
 </div>
 <?php
 }
}
?>

Try using that instead?