Images show up once, then have to clear styles folder to see again?
The fact that derived images can be fetched when they are absent from the filesystem but not when they are present indicates this is a webserver configuration issue.

The webserver rules are normally be set up so that any non-existent paths are routed to Drupal. When image thumbnails are generated they get served by Drupal, but once on disk they should be served directly from the filesystem. Check your .htaccess files (if using Apache) and somewhere there is likely to be an rule denying access to that area of the filesystem.