Equipment/setup recommendations (1 camcorder, 1 mic)
Image quality and audio quality are really two separate beasts, so address each solution independently.  


  Camcorder: Aside from video-related differences, how do video DSLRs or otherwise consumer-grade camcorders compare to professional (big 3-CCD) camcorders in terms of working with external mics?
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"Big CCD" cameras are great at avoiding rolling shutter in the video signal, but this has no bearing whatsoever on the audio they record.  Since your question specifically addresses a stationary scene, though, rolling shutter shouldn't be a problem.  Another big advantage of professional video cameras is their connectivity to broadcast equipment.  But if you're not working in a broadcast environment, these features add unnecessary cost.  

With respect to audio, most dslrs and consumer camcorders ship with some sort of automatic gain compensation (AGC) enabled.  This reduces peak volumes and amplifies silences in an attempt to balance the volume.  In uncontrolled situations, AGC can be helpful, and keep your signal from clipping.  In controlled situations such as interviews, however, AGC will introduce unwanted hiss into the silences.  Make sure the camera you choose allows you to disable AGC for these situations.

Of course, the danger of disabling AGC is that your signal might clip.  To avoid this, consider audio bracketing.  Split the signal from your mic, feed it to a preamp on two channels, and reduce the volume on one of the channels.  That way, if your primary signal clips, you'll have a backup.  Feed both of these channels into your camcorder to avoid sync issues.  

Apologies if this doesn't make sense.  I'm not an audio guy.

In response to your question about lavs vs shotgun mics, I think this is a matter of preference.  Lavs are easy to set up, but they are more conspicuous than shotgun mics.  With a shotgun, you need a dedicated sound man, or a very stationary subject. Keep in mind that you aren't limited to "wired" lavs.  Wireless ones are actually the status quo.