How can I access a node's menu link and weight through EntityFieldQuery
I ended up using db_query JOIN to get the info straight from the database:

$sql = "SELECT menu_links.link_path, menu_links.link_title, menu_links.weight, node.nid 
        FROM menu_links
        JOIN node.nid
        ON menu_links.link_path LIKE CONCAT('node/', node.nid)
        ORDER BY menu_links.weight ASC";
$result = db_query($sql);