Customize SMTP Settings and Email Testing
When altering Admin -> Email Settings -> Customize SMTP Settings, while the toastr will say "Settings saved" or similar, have it actually send an email to the account holder showing that it was successful. That will confirm a lot of things on our side too.
Also, when custom SMTP settings are specified, if something is apparently wrong there, inbound tickets will not actually get created even though the auto responder is broken. So not only do you not actually know that the SMTP settings are not working, you don't even get any sort of indication of a ticket getting created. We don't know on our side, and the sender of course doesn't know either. Just a data black hole.