Just for kicks, I tried doing it at home. The error messages I got were:
"Enter a valid Phone Number" (I had) "Service::Unavailable" ??? "That Code has been used. Hit Refresh Code." (the refresh code link is unresponsive)
and also, the error we got at the Apple Store:
"It looks like the phone number you provided is not registered with iMessage. Please verify the number and try again."
Or rather... thank you for the suggestion. I'm going to dig into the docs on their web page and see if there's a way to contact support w/o having to log in first.
[edit: After a bit of digging around, as best I can tell, I have to log in with an AppleId before I can post a question in the support forums about how to reset my AppleId.]
The first thing support.apple.com asks me to do (when I want to post a question) is to ask me to log in with my AppleId. My problem is that the AppleId that has always worked in the past no longer seems to work and when I try to create a new one, it says my mobile phone number is invalid.