Is the power supply a stock supply?  If not, I'd suggest replacing it; I believe the CC is internally-regulated, so it can take a wide range of input voltages... however, if the internal regulation is crapping out you might have a bigger problem on your hands unless you're handy with a soldering iron.

Can you take a picture of the north-south lines you're talking about?  That might help Duane and/or others troubleshoot exactly what's going on.

As for firmware, I'm a bit fuzzy on the differences between CC & C1, but might it be possible for you to upgrade the firmware via USB?  That may help some of your issues as well; someone else here who has more experience with a CC could probably be of more assistance in this regard.

The volume control issue might be, at least in part, a touchscreen issue. First, switch the device off and clean the screen very gently with a soft damp cloth (no alcohol, no cleaning products). A dirty screen has a noticeable effect on the Chumby's touchscreen performance. Then switch it back on and go to 'Touchscreen' from the Control Panel and follow the calibration procedure.
The Chumby Classic touchscreen is inherently somewhat clumsy and inaccurate and the slider controls do jump around a bit. It is not at all like the touchscreens on new smartphones, and it is as well to bear this in mind.
I think that it is possible that normal variations in component tolerance produce the effect you describe where the volume is right down but there is still sound - the onscreen slider just doesn't match up exactly with the range of the internal component. A similar problem (but in reverse) has been described by many people with the brightness control - the condition of 'zero brightness' occurs before the slider has completely reached the left hand side.
As for your clock issue, the not changing the time manually problem might be a touchscreen thing, might not. As for the other time problem, of my two Classics, one will set its time off the internet straight away; whilst the other takes up to 20 minutes or so before it catches up.

Click on screen touch means that it makes a noise when you touch the screen. From the forum, the general consensus is that the battery lasts about a half hour to a full hour, and is mainly for power glitches. It doesn't harm anything to keep it plugged in. I hope this Chumby newbie answered your questions!

P.S: If you need your Chumbys taken away for a better place, I'll take them for "safekeeping" from your "addiction".

iPod Music only works for older iPod that mount as file systems (not iPod touch, etc).  It tries to read the iTunes database and present a UI to choose playlists.  The code hasn't been rev'd in a while so it may not work except for old stuff.

If someone wants to update the code, it's in the source repo - libipod and chumbipodd.

Oh yeah... I'd forgotten it even did that. I used to use this feature because it was an easier way to get the iPod plugged in to the stereo, since the Chumby was permanently connected.
No, it's not wireless and it's certainly not bluetooth. You needed a cable - I can't remember now, but I suppose it was the standard iPod/USB cable that came with the iPod.

Don't know about the screech, but the Classics' screens do get quite hot.

In particular, what iPod would work with the CC?

I have used Logitech server before but it is a little bit of a hassle since you have to rely on your laptop to feed your chumby. I might as well just listen with my laptop and be done with it.