I receive a "proof failed" error on occasion. I am not positive that this can be controlled based off the notification setting "An error was found that needs my attention" but might be worth checking to see if that's enabled and testing it.
I don't know of an OOTB email notification, but you could build a Document report with the criteria below to view a list of your proofs that have failed to generate.
Note: Sometimes the Proof Status filter can be finnicky and you have to reset it every time you reopen the report to edit (not quite sure why, but a heads up on a bug there)
