In other words, the volume of the music audio has to
be fairly high, or the Muse will cut out during softer parts of a song.

The audio output device needs a compression circuit to keep audio levels
more near a constant output. There are portable cd players out there that
have such a feature.


Another issue is that my unit only seems to pass one of the stereo
channels through to the headset. I hear the music in both ears, but it's
only the left or right (not sure which) channel.


Your headset may have a dip switch that's factory set to mono-out a stereo
input - like the Bose headset. Try the other setting. Or it may be wired in
mono. Try it on a stereo source at home.