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.

Short answer

Hangouts will display the "official" name in Google+ or in About Me if you are trying the new Google+. Users have several settings in order to choose how it will be displayed.

Details

"Old" Google+

To change how your own name is displayed in Hangouts,

1. Go to http://plus.google.com/me
2. Click on the name displayed in the top banner.
3. Click over the "+More options" link
4. Edit the name fields "at will"
5. On "Display my name as", select the desired option
6. Click over the Save button.

New Google+

1. Go to http://aboutme.google.com
2. Click on the button.
3. Edit the name fields "at will"
4. Click over the OK button.

References

From Change the name on your Google Account - Google Accounts Help

Change the name on your Google Account

You can change the name on your Google Account by editing your Google+ profile. Editing your profile name in Google+ will change your name in all the Google products you use for this account. For example, you'd see your name change in Gmail if you send emails using your Google Account name and YouTube if you’ve linked your YouTube channel with your Google+ profile.

You can also edit other profile details such as your birthday or work experience, and control how you share that information by changing your Google+ profile settings.