[jrblasde]

Alright, I feel like a dummy at this point. I moved the transformer to a different surface and plugged it back in. This time no rattle. I took it back to the first table, and the rattle resumed. Turns out that the magnetic field had been rattling a little piece of metal that I didn't see at first!  

So now there is no humming sound, and voltages look good! I left it plugged in for five minutes, and it did not overheat. Not even warm to the touch. So I have a usable transformer!!!

I'd like to thank everyone for their guidance on this transformer restoration. I had absolutely no idea that these could be cleaned up and the wires sleeved.