It doesn't sound like any of you are having the same problem, unless you're all forgetting to mention you're running multi-site. I'd appreciate it if you started your own support threads because your issues don't seem to be related to mine.
I'm still getting the same problem when trying to minify, however now I'm getting a more useful error message:
Minify Auto does not work properly. Try using Minify Manual instead or try another Minify cache method. You can also try a lower filename length value manually on settings page
This gives me a link to a settings page, but I then get a permission denied message. I presume this is because I'm using multi-site and the plugin is in community mode as the link directs to the W3TC settings for that particular site. Anyway, if I manually go to the Minify settings and reduce the file name length to something shorter, like 25, I still get the same error.