Add search icon to footer navigation

And to what direction the search-input field should open?
Where does the ajax search results should toggle?
What happend to the scroll to top button?

This in child-theme functions.php will place the magnifier icon in the menu3 of enfold:

function avia_append_search_nav3 ( $items, $args )
{
    if ((is_object($args) && $args->theme_location == 'avia3') || (is_string($args) && $args = "fallback_menu"))
      {
          global $avia_config;
          ob_start();
          get_search_form();
          $form =  htmlspecialchars(ob_get_clean()) ;

          $items .= '<li id="menu-item-search3" class="noMobile menu-item menu-item-search-dropdown menu-item-avia-special">
                 <a href="?s=" data-avia-search-tooltip="'.$form.'" '.av_icon_string('search').'>
                  <span class="avia_hidden_link_text">'.__('Search','avia_framework').'</span>
                 </a>
               </li>';
      }
      return $items;
}
add_filter( 'wp_nav_menu_items', 'avia_append_search_nav3', 10, 2 );

but i do not know how to get the ajax search results to show

ok – we posted the same time.
How did you create that footer? is it a page as footer ?

Well a quick and dirty way would be to activate the main menu search icon ( Dashboard – Enfold Child – Main Menu: “Append search icon to main menu”)
– and shift this to your other menu:

function shift_search_icon(){
?>
<script type="text/javascript">
(function($){
	$('#avia-menu').find('#menu-item-search').detach().appendTo('#menu-menu3');
})(jQuery);
</script>
<?php
}
add_action('wp_footer', 'shift_search_icon');

on my test page is stayes at the menu.
can you try this:

function shift_search_icon(){
?>
<script type="text/javascript">
(function($){
	$('#avia-menu').find('#menu-item-search').clone().appendTo('#menu-menu3');
         $('#avia-menu').find('#menu-item-search').remove();
})(jQuery);
</script>
<?php
}
add_action('wp_footer', 'shift_search_icon');

just to see if detach() is the problem

Yes – it is the way you style your menu to have not the treestructure there but besides each other.

Look how you reach it in your css and try to substitute it by:

#footer .widget {
    padding: 0;
    margin: 30px 0 30px 0;
    overflow: visible;
}

#footer .widget_nav_menu ul {
    display: flex;
    flex-flow: row wrap;
    justify-content: center;
}

#footer .avia-search-tooltip {
    top: 100% !important;
    margin-left: -130px;
}

but i do not know why the ajax search proposals aren’t there – did you switch of that feature?