The old administrator has left the company so I went into the control panel, removed his admin rights, changed his password, forwarded the inbound messages to another user, and changed the email address of the primary administrator from him to me, but Quarantine summaries and other system messages still have his email address in the From: field, even two weeks later.

How can I stop that happening?

I also changed the Secondary email address in case I lose access and need a password reset sent to a backup account, but if the Primary address change didn't take how do I know the Secondary address works? What if gmail sends the password reset message to the old administrator's private off-domain email account? I might get locked out of my own domain.

This needs fixing immediately, your forums are full of gripes about this topic since 2009, what is going on Google? Get off your scooters and bean bags, stop playing pool and do some work!