How do I know if a rewritten rule was applied?

I’m not very sure of what you’re trying to achieve, but I hope this answer can help. 🙂

1. With the way your custom post type is being registered, WordPress will create custom rewrite rules such as below for the CPT’s single post pages:

   RegEx: ballinfo/([^/]+)(?:/([0-9]+))?/?$
   Query: index.php?balls=$matches[1]&page=$matches[2]
   

2. So for the following question:

   how do I know if a site was called with /ballinfo or not

   When the single page of a balls post is visited, the URL would have /ballinfo/ as in example.com/ballinfo/an-example-balls-post.

   Now to programmatically check if the URL contains /ballinfo/, you can check if the $request property of the global WP class instance starts with a ballinfo/ like so:

   global $wp;
   if ( preg_match( '#^ballinfo/#', $wp->request ) ) {
       echo 'Site was called with the /ballinfo<br>';
   }
   echo '$wp->request is ' . $wp->request . '<br>';
   

3. And for the following question:

   How do I know if my rewrite rule was matched and used?

   You can match the rule (RegEx) with the one in the $matched_rule property of the global WP class instance.

   For example, for the single balls post pages, where the rule uses the RegEx pattern as in point #1 in this answer, try this:

   global $wp;
   if ( 'ballinfo/([^/]+)(?:/([0-9]+))?/?$' === $wp->matched_rule ) {
       echo 'Yay, my rewrite rule was matched!<br>';
   } else {
       echo 'Not matched. $wp->matched_rule is ' . $wp->matched_rule . '<br>';
   }
   

And you may already know this, but if you just wanted to check if the requested URL is for a CPT post/archive/etc., you can use is_singular(), is_post_type_archive(), and other appropriate WordPress conditional functions/tags.