combining log and error output
I can think of a way to do it the other way around, by defining your own civicrm logger and having it send to syslog. The problem is large stacktraces might get cut off and be formatted badly because of the way that works.
To get php to send all its errors including 500-level errors to civi might not even be possible, because they happen before civi even gets to the point where it would be capable of writing a log. If civi can't even load because of error 500, it can't possibly write to its own log.
Or if the error were to be in the CMS, then it might not be able to get to civi either.