How can I use the administration theme for the user/%/edit page? How can I get the user/%/edit page to use the admin theme that is used for the rest of the site?
That path is already one that will use the admin theme if the current user has the correct permissions to access it.

Try going to http://mysite.com/admin/people/permissions and checking the "View the administration theme" permission for the roles you wish to be able to use the admin theme.

It'll definitely work as the User module implements hook_admin_paths() to define that path as an admin page:

function user_admin_paths() {
  $paths = array(
    'user/*/cancel' =&gt; TRUE, 
    'user/*/edit' =&gt; TRUE, 
    'user/*/edit/*' =&gt; TRUE,
  );
  return $paths;
}