Hi,
Just wanted to update this thread with the results of my misadventures.
I sent the unit back to Apogee, and they sent it back to me, and it still had the same problem.
Apogee was VERY COOL about this, and I thank them for it. They worked with me and the Guitar Center (also VERY COOL) where I bought it, to arrange for me to get store credit.
So I used the store credit to get an RME Fireface 800. I've just now got the unit, and it seems to be working fine. The RMEs are noted for their reliability, from what I've read, so I think my interface issue is solved.
So to those who are having similar problems (and I still have no idea what the cause was)you should contact Apogee. They were extremely honorable in helping me with this issue, and I can happily recommend them, understanding that if you have any issues, they will not be jerks about it 🙂
Thanks Apogee, and thanks Guitar Center. Maybe in a year or two, I'll check out some more Apogee products, because, other than the hiss, the Duet did sound pretty good.