Robin

i am not sure if i follow what exactly you want but what i can understand is that when a user clicks on a link on ur page you want that page to get an overlay while the user goes to another site, if thats the case
you could add a div right after body with an absolute class element having dark background color with low opacity and width of your container and that class is by default hidden and through js
class.onClick ( make absolute class display block)

If you have your landing page as index.html/index.htm (you could just change that in your hosting panel to allow first to render html pages and than php pages)
this way wordpress index.php wont render – and only ur landing page will work, (but still if u need to run your WP u can put that into a sub folder)

this is working on my local try it out

global $wp_query;

	$big = 999999999; // need an unlikely integer

	$links = paginate_links( array(
		'base' => str_replace( $big, '%#%', esc_url( get_pagenum_link( $big ) ) ),
		'format' => '?paged=%#%',
		'current' => max( 1, get_query_var('paged') ),
		'total' => $wp_query->max_num_pages,
		'prev_text' => '<i class="fa fa-left"></i>',
		'next_text' => '<i class="fa fa-right"></i>'
	) );
?>
	<div>
		<?php echo $links; ?>
	</div>

u know the link you have posted are of localhost and not actual server + u should open a new post for your issue

Your NameServer and DNS entries would go in ur domain control panel – you should contact your hosting, This is more of WordPress support forum