How to eliminate noise on iPad analog/headphone output?
Well to my shock and amazement, I seem to have solved the problem on my own.  Looks like it was actually related to the specific USB wiring of my setup.

I have a 10-port USB Hub (un-powered as I lost the AC adapter at some point) with most of the ports filled (a USB mic, a couple pieces of external MIDI gear, iPhone, Mac keyboard, etc.)  In addition to the devices I just mentioned, I also had my audio interface connected to this hub, but had forgotten that it was so.  The iPad mini was connected directly to the mac.

By switching things around and connecting the audio interface directly to the mac instead of to the hub, all extraneous noise disappeared completely.  Everything is pristinely quiet now (as best as I can tell at the moment, being that this is an active household and it's not exactly silent very often).  Definitely the constant whirring and chirping and data noises I was hearing before, are gone.  As is the high pitched whine.  Interestingly the iPad works fine connected to either the Mac or the hub, however does not get power if connected to the hub, so I'm leaving it on the mac.

Just goes to show how critical the way things are plugged in can be.  Just because it's a USB port, does not always mean it's the right one to plug into!  =)