Making some fields required and others not

This can be accomplished with a custom ctools format plugin.

YOUR_MODULE/plugins/format/address_custom_required.inc

/**
 * Requires Street and City but not ZIP.
 */

$plugin = array(
  'title' => t('Street and City are required, ZIP is optional'),
  'format callback' => 'addressfield_alter_format_address_custom_required',
  'type' => 'address',
  'weight' => 100,
);

/**
 * Format callback.
 *
 * @see CALLBACK_addressfield_format_callback()
 */
function addressfield_alter_format_address_custom_required(&$format, $address, $context = array()) {
  // Require Street address
  $format['street_block']['thoroughfare']['#required'] = TRUE;
  // Require City
  $format['locality_block']['locality']['#required'] = TRUE;
  // Make ZIP Optional
  $format['locality_block']['postal_code']['#required'] = FALSE;
}

This will need to be placed in a folder that ctools has been told is a format plugin directory. You can accomplish that with the following in your YOUR_MODULE/YOUR_MODULE.module

function HOOK_ctools_plugin_directory($module, $plugin) {
  if ($module == 'addressfield') {
    return 'plugins/' . $plugin;
  }
}

/**
 * Implements hook_ctools_plugin_type().
 */
function HOOK_ctools_plugin_type() {
  $plugins['format'] = array(
    'load themes' => TRUE,
  );
  return $plugins;
}

You can adjust the weight as needed to get it to apply in the correct order relative to the other formatters.

Thank you, RedEight!
Feel free to copy your answer to a new documentation page (click "add child page" on https://www.drupal.org/node/1267280), for extra karma ;)