How to call watchdog() outside Drupal
Requiring the existence of a Drupal installation for all of your other non-Drupal PHP projects isn't something I've come across before...but you can bootstrap Drupal to make the watchdog() function available like so:

define('DRUPAL_ROOT', '/path/to/drupal/installation');
include_once(DRUPAL_ROOT . '/includes/bootstrap.inc');
drupal_bootstrap(DRUPAL_BOOTSTRAP_FULL);

watchdog('my_module', 'A message');