Slow query?

Yes: don’t use Relevanssi for queries that don’t have a search term. That should be disabled by default, so don’t enable it. As you’ve seen, it’s heavy, and the benefits of using Relevanssi when there’s no search term are fairly minor.

Ah, sorry, my mistake, that only applies to admin searches. Add this to your site (in theme functions.php, for example):

add_filter( 'relevanssi_search_ok', function( $ok, $query ) {
    if ( empty( $query->query_vars['s'] ) ) {
        $ok = false;
    }
    return $ok;
}, 10, 2 );

This will disable Relevanssi in termless searches.

No, the plugins are unrelated. Relevanssi Light ignores Relevanssi.

Have you made changes that require the processing? In general, it’s not necessary. Your database does the indexing automatically, you don’t have to do anything.

Yes, it needs an index, but your MySQL database server does that automatically. It’ll take a while to get it running, but it should be pretty fast.

You only need to process the posts if you want to include custom fields or do some kind of custom data inclusion. Otherwise, the plugin is 100% automatic with no user action required.