UPDATE!

I did all the things we were talking about. I switched back to the original chain to see if the problem persisted, which it did. I took another link out of the new chain, which helped with the shifting, but didn't solve the rattling problem. Upon close inspection, (which means I took my bifocals off and stuck my face down right next to the derailleur), I determined that the two pulleys were not in alignment. I bent the derailleur cage by hand, and that seemed to do away with most of the noise. My hunch is that the poor derailleur got slowly bent after sitting in the parts bin for twenty years. I don't think that it's the derailleur hanger, since I didn't have any problems with the previous derailleur, a Suntour Superbe.

Thanks all for your help. I'll update again if the situation warrants.