This was painful, but everything works now. I had to also completely uninstall iClone and then repopulate the Template folder, so this was a lot of work. Unfortunately, I don't know what was wrong exactly, which would have been useful to prevent this from happening in the future.
This comment is really associated with #7745, so I would like for this entry to be kept open. What I had not mentioned, because it didn't seem important, that I had renamed the iClone 7 Template folder in preparation for a complete reinstall of iClone 7. So during the reinstallation of CC, there was technically no iClone 7 Template folder. After I had determined that CC worked, I thought I would try iClone without reinstalling. So I restored the Template folder name and launched iClone, but it got stuck on the Prime Hair styles folder. I then tried CC, but to my dismay it also got stuck, whereas before it was OK, as reported. So I then tried to rename the iClone Template folder (with CC open), but got a message that it was open by some application, which could only be CC. So I closed CC, and could then rename the folder. After that, I launched CC and could again access all Hair folders. I will try my original plan to reinstall iClone 7 and see what happens. It seems though that some data sharing is going on between CC and iClone 7 (I think it is the Legacy hair) that causes problems in some cases.