Not the OP, but I have a similar setup, however instead of Postfix I am using OpenSMTPd, which I found much easier to configure.

Checking for false positives is a manual process. I've never had legitimate mail marked as spam though, only the other way round.

My experience with Gmail's filter is not as nice btw. At work, I had a couple of situations where important mails from a customer were flagged as Spam, which went unnoticed for two weeks.

Pretty much. I just check it occasionally. Did the same thing back when I was a Gmail user, because Gmail has false positives too from time to time.