Is there a module or configuration that logs PHP mail errors? On shared hosting. We're having strange mail problems... mails do not get sent to various domains or are delayed. It seems to happen more in Webforms than in Notify_message actions in Rules.

Is there a way to log the PHP mail() error messages without hacking the MimeMail module? The Drupal log only shows that there was an error... nothing more specific.
Have a look at maillog and Mail Logger. They both log all mail activity handled via drupal_mail. See what they have to offer as far as mail errors go.