Setting $cookie_domain variable for specific folder structure

The short answer is that you cannot set cookie domain to such a value. See http://www.php.net/manual/en/session.configuration.php#ini.session.cookie-domain

As a workaround, you can use this in settings.php (around where $cookie_domain is set:

$cookie_domain = ".example.com";
// Set cookie_path ourselves as Drupal doesn't.
ini_set('session.cookie_path', '/mynewsitefolder/');

$cookie_domain is eventually used to set session.cookie_domain but there is nothing in Drupal code base to set session.cookie_path and so we directly set that in settings.php itself.