How do you get the display settings of a field on a particular content type/display
You're looking for field_info_instance()


  Returns an array of instance data for a specific field and bundle


Example usage:

$info = field_info_instance('node', 'field_foo', 'article');
$display_settings = $info['display'];


$display_settings will be an array of display settings keyed by the view modes that are customisable for that content type and bundle.