Okay... I was able to talk to Quill? Quinn? at Apple Support who stepped me through the process of recovering an AppleId from a gmail account I no longer use. The key here was to remember the security questions, which, thankfully, I did. I don't know if there would have been a way to recover it had I forgotten them. Once I proved to the system I was the "real" owner of the AppleId at that account, I was able to add my phone number and change the security questions. Not sure why I wanted to do that last step, but it seems like I should change them (and then write down the answers on a piece of paper I put in a safe deposit box.)
The weird thing is I think I did all these things before. I did identify myself to the Support Rep (Quill? Quinn?) -- did they put this one account back in a state where it would allow me to reset? The world is filled with mystery.
Next we're going to see if I can apply for the position I originally wanted to apply for using this AppleId.