Make table disappear when fields are left blank?

Probably the easiest thing to do would be to fetch the fields above your table, and assign them to variables:

$anime_anname = get_post_meta($post->ID, 'anime_anname', true);
$anime_angenre = get_post_meta($post->ID, 'anime_angenre', true);
...

Then just wrap the table in an if statement that looks like the following:

if (
    !empty($anime_anname)
    || !empty($anime_angenre)
    || ...
) {
// Table goes here.
}

Of course within your table, be sure to use the variables as well, rather than fetching the feeds again.

Hope that this helps.