Replace
if ( $comments = $wpdb->get_results("SELECT comment_author, comment_author_url, comment_ID, comment_post_ID FROM $wpdb->comments ORDER BY comment_date_gmt DESC LIMIT 5") )
with
if ( $comments = $wpdb->get_results("SELECT comment_author, comment_author_url, comment_ID, comment_post_ID FROM $wpdb->comments WHERE $wpdb->comments.comment_approved = '1' ORDER BY comment_date_gmt DESC LIMIT 5") )
This will get you to display approved comments including trackbacks and pingbacks. Should you wish to have only comments listed, use the following.
if ( $comments = $wpdb->get_results("SELECT comment_author, comment_author_url, comment_ID, comment_post_ID FROM $wpdb->comments WHERE $wpdb->comments.comment_approved = '1' AND $wpdb->comments.comment_type = '' ORDER BY comment_date_gmt DESC LIMIT 5") )