This is mainly intended for the plugin's author, who has most certainly created the most useful WordPress plugin out there. Hopefully I can help making it work again.
In short, minify doesn't work; regardless if I use auto or manual. With auto I get 'wrong param" for both CSS and JS when trying to download the files. Also, it doesn't support CDN (I use Cloudfront).
With manual, there seems to be two problems:
1. A minified CSS-file is produced to wp-content/w3tc/min/minify_5f67...css as well as files wp-content/w3tc/min/8a12d/default.include.css.id and wp-content/w3tc/min/8a12d/default.include.js.id. Both the default.include... files only contain a key/value pairs 'i:12...' and 'i:18...' respectively.
2. No minified JS-file is produced at all.
Regarding CDN, the minified CSS-file is correctly uploaded to Cloudfront. However, in the generate HTML, the following two CSS and JS URLs are given (on the CDN): wp-content/w3tc/min/8a12d/default.include.12...css.gzip and wp-content/w3tc/min/8a12d/default.include.18...js.gzip. Both don't exist, and if followed only return a "NoSuchKey The specified key does not exist" errors. The site is thus without CSS and JS.
Further settings that I have seen mentioned: I have tried with both URL rewriting on and off; various minification algorithms; CDN on and off; and of course, minify auto and manual.
I'm happy to provide further details if that helps.