If there is no WP in public root, create a flat index.php or index.html file there that looks the way you want, including links to elsewhere. This becomes the entry page for your primary domain.

The logos and other graphics do not need to be backgrounds. You can link any image to where ever with simple img tags wrapped in an anchor tag. One cannot click on backgrounds to follow links, there has to be an object to click on, even if it’s a transparent overlay.