I think the trick was to open pavucontrol, find the GF108 HD Audio output, and then unmute it by clicking the mute toggle button. I also muted the built-in audio analog output.
In System Settings -> Multimedia -> A&V, Audio Hardware Setup tab, select GF108 and the profile (HDMI 4).
In the Device Preference tab, I also played around with moving the GF108 up in the preference priority for the Video and Communication categories, but after some more experimentation I'm not sure that made any difference.
Google Chrome sometimes cannot open the USB video camera. It seems to help to just keep selecting the video device in the camera dropdown in the Hangout Settings dialog. Eventually Chrome opens it. Maybe restarting Chrome or unplugging the USB video camera and plugging it back in would help also.
In the Hangout Settings dialog, make sure the audio output is set to the Default, not to the built-in audio output or to the HDMI output.