Well, good. I got a reasonable response from the university's help desk, saying that I can stop by and export all the stuff that I lost in their migration to the new system. Also, it turns out that I'm eligible for Microsoft Exchange (which is the system they're using for staff and faculty) and can migrate over to that. I wish that we'd been better informed about this switch-over: the information I found made it sound like I would only get access to Exchange during semesters when I'm teaching, and the idea of switching back and forth is just annoying.
The help desk guy also asked me for specific feedback regarding my problems with the school's webmail system, and I gave him a list. And now, for the first time, I have my university email set up with Windows Mail (because I have my personal email set up with Thunderbird, and it's annoying to switch back and forth between two accounts that you want to keep separate), and so the problems that I have with webmail will only be annoying when I am away from my computer.
I am not in Hulk Smash mode now. That is good.