Then I tried downloading the WoW trial version just to see if it was a problem with Blizzard or my actual computer. Turns out I can't download the latest patch for WoW either, so it's definitely a design problem on Blizzard's part.
Typical Blizzard. Instead of trolling us with release dates, they create an open beta that nobody can actually download.
Just thought I'd leave up my error message numbers so anyone with a similar problem can know it's not their machines fault.