Cannot install the module

Thinking about this again - I'm not sure that doing the directory creation in hook_requirements on install is correct. I think the directory creation should happen in google_tag_enable() as hook_requirements runs before a module is installed.

That said the fix in #2 looks like it will work.

Note all you have to do is make the requirement hook do nothing on install because \GTMContainerManager::createAssets() is already called from google_tag_enable() and that will create the directory if necessary - and also output messages if it fails.

Also I'd change
if ($phase == 'runtime' || $phase == 'update' || $phase == 'install') {
to
if ($phase == 'runtime') {

There's no need for this to prevent updating your site if it fails.

Which means you could change the requirements hook to:

/**
 * Implements hook_requirements().
 */
function google_tag_requirements($phase) {
  // This hook only makes runtime checks.
  if ($phase !== 'runtime') {
    return array();
  }

  $requirements = array();
  $manager = \GTMContainerManager::getInstance();
  $containers = $manager->loadContainers();
  if (empty($containers)) {
    // Google Tag Manager container ID has not been set.
    $args = array(
      '@url1' => '/admin/config/system/google_tag/settings',
      '@url2' => '/admin/config/system/google_tag',
    );
    $description = t('Configure default settings on the <a href="@url1">module settings page</a>. Afterwards, add a container on the <a href="@url2">container management page</a>.', $args);
    $requirements['google_tag'] = array(
      'title' => t('Google Tag Manager'),
      'description' => $description,
      'severity' => REQUIREMENT_WARNING,
      'value' => t('Not configured'),
    );
  }

  // Adapted from system_requirements().
  $directory = \GTMSettings::getInstance()->get('uri');
  if (empty($directory)) {
    $requirements['google_tag_snippet_parent_directory'] = array(
      'title' => t('Google Tag Manager'),
      'description' => t('The snippet parent directory is not set. Configure default settings on the <a href="@url1">module settings page</a>.', array('@url1' => '/admin/config/system/google_tag/settings')),
      'severity' => REQUIREMENT_ERROR,
      'value' => t('Not configured'),
    );
    return $requirements;
  }
  $directory .= '/google_tag';
  if (!is_dir($directory) || !_google_tag_is_writable($directory) || !_google_tag_is_executable($directory)) {
    _file_prepare_directory($directory, FILE_CREATE_DIRECTORY | FILE_MODIFY_PERMISSIONS);
  }
  $is_executable = _google_tag_is_executable($directory);
  $is_writable = _google_tag_is_writable($directory);
  $is_directory = is_dir($directory);
  if (!$is_executable || !$is_writable || !$is_directory) {
    // The snippet directory does not exist or is not writable or searchable.

    // If applicable, get the directory path of stream wrapper.
    $wrapper = file_stream_wrapper_get_instance_by_uri($directory);
    if (method_exists($wrapper, 'getDirectoryPath') && ($path = $wrapper->getDirectoryPath())) {
      // getDirectoryPath() is not defined in StreamWrapperInterface; it
      // exists in LocalStream and the local storage replacement classes in
      // google_appengine; s3fs returns an empty string.
      $path .= '/google_tag';
    }
    elseif (!($path = $wrapper->getExternalUrl())) {
      $path = $directory;
    }

    if (!$is_directory) {
      $error = t('The directory %directory does not exist.', array('%directory' => $path));
      $description = t('An automated attempt to create the directory failed, possibly due to a permissions problem. Create the directory and make it writable.');
      $value = t('Does not exist');
    }
    elseif (!$is_writable) {
      $error = t('The directory %directory is not writable.', array('%directory' => $path));
      $description = t('An automated attempt to make the directory writable failed, possibly due to a permissions problem. Make the directory writable.');
      $value = t('Not writable');
    }
    else {
      $error = t('The directory %directory is not searchable.', array('%directory' => $path));
      $description = t('An automated attempt to make the directory searchable failed, possibly due to a permissions problem. Make the directory searchable.');
      $value = t('Not searchable');
    }
    if ($phase == 'install') {
      $description .= t(' For more information, see INSTALL.txt or the <a href="!handbook_url">online handbook</a>.', array('!handbook_url' => 'https://www.drupal.org/server-permissions'));
      $value = '';
    }
    $requirements['google_tag_snippet_directory'] = array(
      'title' => t('Google Tag Manager snippet directory'),
      'title' => t('Google Tag Manager snippet directory'),
      'description' => "$error $description",
      'severity' => REQUIREMENT_ERROR,
      'value' => $value,
    );
  }
  return $requirements;
}

As there's less conditionals and indentation and the intentions of the implementation are easier to grok. You can also lose the $t = get_t() as that is not necessary.