Would like to swap the search function…

I can not tell you how to integrate the login form itself – the code depends on your application. However the search form code can be found in functions-enfold.php

/* AJAX SEARCH */

if(!function_exists('avia_append_search_nav'))

{

//first append search item to main menu

add_filter( 'wp_nav_menu_items', 'avia_append_search_nav', 10, 2 );



function avia_append_search_nav ( $items, $args )

{

if ($args->theme_location == 'avia')

{

global $avia_config;

ob_start();

get_search_form();

$form =  htmlspecialchars(ob_get_clean()) ;



$items .= '<li id="menu-item-search" class="noMobile menu-item menu-item-search-dropdown"><a href="?s=" data-avia-search-tooltip="'.$form.'">'.$avia_config['font_icons']['search'].'</a></li>';

}

return $items;

}

}

The first code block:

ob_start();

get_search_form();

$form =  htmlspecialchars(ob_get_clean()) ;

generates the form and stores it into the $form variable, the second code block

$items .= '<li id="menu-item-search" class="noMobile menu-item menu-item-search-dropdown"><a href="?s=" data-avia-search-tooltip="'.$form.'">'.$avia_config['font_icons']['search'].'</a></li>';

adds the search icon ($avia_config) to the main menu and adds the $form variable to the tooltip content.