The issue was that because this server is a "Managed Service" with our host, they installed a new system for managing rules for Mod_Security (an Apache server module aimed at stopping common attacks like SQL Injections and the like).
This new system was incompatible with the old configuration format we had, which had certain rules disabled (e.g. the rule that normally triggers a blank page when you add your URL to your support query, due to believing it's a remote inclusion attack).
I have been watching the server's error log, and the issues should now be resolved.