Strange Page not Found, Permissions, Clear cache
An answer to part of your question:

Then I noticed the strangest thing, that whenever I would clear the
cache at, I would would get an entry in the drupal_debug.txt
it entered in hook menu
I do not get this entry in the drupal_debug.txt when accessing the
wrong-answer page, I only get it when clearing the cache.

If Drupal built the menu for every page load, it would result in heavier page loads. As such, hook_menu() is only called after the cache is cleared, as the menu cache is rebuilt the first time it is needed and no cache exists. Once it's been cached, hook_menu() does not get called anymore.