Google Hangouts – Ignore ‘Joined the Hangout’ Notifications

From Facebook's point of view, this is a feature, not a bug. FB wants group admins to deal with members ASAP. Your main options are:

(edit: open groups have the same approval issues as closed ones)

1. change it to a page so people can join without approval.
2. change it to a secret group so people can't request to join.
3. give up your admin role and have someone else deal with it.

See also: Any way to auto-approve requests to join new-style Facebook group?

It appears that chat/hangout messages are auto-archived in Gmail afterall - providing you don't have any filters set on the user you are chatting with! (This sounds like a bug if you ask me!)

I previously had a filter set on this particular user that simply prevented any emails from that user being seen as spam - but this was apparently enough to prevent all hangout messages from that user being archived. I had, however, already deleted this filter - but it appears to have taken some time for the system to catch up, since I was still getting hangout messages in my inbox after I had initially deleted the filter.

Thanks to Grzegorz for help in diagnosing this.