I did it! I fixed my PDF problem.
For a while, whenever I tried to open a PDF, Adobe Acrobat Reader would start, but then it would just hang there forever.
I had basically just given up, but then today I accidentally clicked on a PDF file online (or well, the clicking wasn't an accident, but I forgot to check first to make sure it wasn't a PDF because if I'd noticed that, then I would have known not to click it). Anyway, it launched Adobe Acrobat, crashing my browser and severely slowing down the computer.
I forced Adobe to end, and as it was closing I caught a glimpse of an error window saying something about "font capture". So I googled "troubleshoot adobe font capture", found this article by someone who'd fixed exactly the same problem I was having after he happened to come across this support document at Adobe. The problem is that it tries to load up all the Temp files on your system, and for some reason, the Temp files never disappear, they just accumulate. They don't sound very Temp to me... And I have no idea what this has to do with Font Capture except that maybe that's as far as Adobe ever gets when it tries to load up? Who knows.
Anyway, deleting all my temp files completely fixed the problem. Except that from home I was only able to download the free reader from the web and not the full Adobe Acrobat professional, but I'll get that from school at some point. For now though, I can finally access all the PDFs stored on this computer. It's so, so nice.
It's kind of funny, though, how I arrived at the fix. For all my googling earlier trying to troubleshoot the problem and our IT guy also trying to figure it out, we couldn't find the problem. But then I just happen to notice something about "font capture" which isn't actually the problem I'm having, but it leads me to a page that leads me to another page. I guess that's the nature of research, but I'm left with this feeling that it was just fortuitous luck that got me to this point and that I will continue to have no idea how to fix these sorts of problems in the future... At least now if any of you ever have this problem, you'll have a possible solution at your fingertips.