Use search.php for custom search results? Also wp query pagination not working

I am making a filter plugin and I want to use the search.php for the results, but if I set text input name to ‘s’ in my plugin, then it uses the regular search functionality, not the WP_Query method I set up in my plugin.

And if I change the name and make the plugin use that name, I get the results but in the same place where the plugin itself appear.

So, can I somehow make the search.php page use the results from my plugin?

the form and my query:

<?php

    $blog_url = get_bloginfo('url');

    $form = <<<EOH
    <div id="sbc">
    <form method="get" action="{$blog_url}" id="ss-search">
        <input type="text" value="{$search_text}" name="ss" onblur="if (this.value == '') { this.value = '{$search_text}';}" onfocus=if (this.value == '{$search_text}') { this.value = '';}" />
        {$list}
        <input type="submit" id="sbc-submit" value="Search" />
    </form>
    </div>
EOH;

if (isset($_GET['ss'])) {
        $args = array(
            // 'category__not_in' => 1,
            's' => $_GET['ss']
        );

        $q = new WP_Query($args);

        if ( $q->have_posts() ) {
                echo '<ul>';
            while ( $q->have_posts() ) {
                $q->the_post();
                echo '<li>' . get_the_title() . '</li>';
            }
                echo '</ul>';
                echo get_pagination_links();
        } else {
            echo 'no posts found';

        }
        /* Restore original Post Data */
        wp_reset_postdata();
    }

    echo $form;

?>

search.php

<p class="search_results">Your search for <span class="searchterm"><?php the_search_query(); ?></span> has returned <?php echo $totalresults; ?> results.</p>

<?php

global $paged;
if(empty($paged)) $paged = 1;

$loop_counter = 1;

$results_per_page = get_query_var('posts_per_page');

echo '<ul class="search-list">';

if ( have_posts() ) : while ( have_posts() ) : the_post();

  if( $paged == 1 ) {
    $real_count = $loop_counter;
  } else {
    $real_count = $loop_counter + ( $paged * $results_per_page - $results_per_page);
  }

  echo '<li><span class="listnum">' . $real_count . '.</span>';

  // The Post
  ?>
    <a href="<?php the_permalink(); ?>"><?php echo substr(strip_tags($post->post_content), 0, 46);?>...</a> 

  <?php

  echo '</li>';

  // genesis_after_post();

  $loop_counter++;

  endwhile;
  // genesis_after_endwhile();

else :
  // genesis_loop_else();
    echo '<p>No results found. Search for "page" for best results.';
endif;

echo '</ul>';

?>

Also, pagination doesn’t work witht the custom WP_Query ($q). I DO get pagination links, but they don’t work. For example if I click on the second page, nothing happens. If I change $q to $wp_query it works, but then I can’t pass it the arguments I want.