Following a fresh install of Debian 7.7, I have no sound. No media players, system sounds, even the test in systemsettings yields mere silence. The default configuration is phonon-vlc-backend with pulseaudio. Alsa as well. I am using a nvidia HDMI sound card (which is detected). Oddly, when I installed the same system but with GNOME, sound worked quite well. GNOME uses gstreamer backend and so I thought to simply install that in my KDE box, but that did not solve the problem. I am fairly new at this, so I don't know what data is necessary to begin the investigation.

Note-- in desperation, I purged pulseaudio and that yielded *some* sound in the systemsettings test, but that sound was not present anywhere else in the system and I've since done a fresh reinstall. Back at square one.

When making an out of the box installation there should be sound, if that is not the case, then something went wrong with your installation or you have a very uncommon hardware.

FWIW: purging pulseaudio is certainly NOT the solution, the first step would be to check with alsamixer if there are muted channels.
FWIW2: GNOME does not use Phonon or any of the Phonon backends, so you are comparing apples with oranges here...

I have audio with the line out (headphone jack) but still no luck with HDMI.

I doubt you will ever have sound with HDMI if you don't use pulseaudio, removing that is not recommended...