Update: As noted by ss1 and DFrost above (Thanks!), we got the fix rolled out onto our public servers about an hour ago. Please let us know here if anyone continues to experience problems.
I believe the final diagnosis was that we were having problems with the Terms of Service agreement system. People who had not agreed to the TOS (new users) or people who needed to agree to the most recent TOS were being blocked from downloading and experiencing other oddities, but were not being presented with the little window asking them to agree to the Terms of Service. We got the TOS agreement system back up and running smoothly. I think everyone who was blocked will be prompted to agree to the Terms of Service, then they will be able to proceed on to the download.
Thanks for your patience and help in troubleshooting this nasty issue.
Chris Fullmer