How can I send an e-mail whenever user registration fails?
Not sure how deep you want to go, but here's what I would do.


Create a custom module, "mysite_utils" (I always make at least one
custom module to do some custom tweaks.)
Create a mysite_utils_form_alter(&amp;$form, &amp;$form_state, $form_id)
function.
In that form_alter function, look for the user registration form_id.
Add a custom validation handler mysite_utils_user_register_validate().
Put your validation checks in that function. If the validation fails,
send yourself an email.


This process will require a lot of custom coding, not sure what your Drupal level is.