How can I take control of the rendering of a file entity when inlined in text?
Your best bet is http://drupal.org/sandbox/DevinCarlson/1823634 - which is an add on to media, but I'm not certain whether or not it can also display fields that are attached to a file entity.

As per http://drupal.org/node/1792738 this functionality is planned to be rolled into the media module at some stage.