@ M One
If you're on a server with CPanel installed, chances are W3TC will not be able to tell you whether or not the server you on has the modules required installed. I know this is the case with mod_deflate and mod_headers as it's always shown Not Detected on every installation I've done, even though they are installed and working just fine. In such a case, you'd need to get in touch with your web hosting provider to find out for sure.
As for testing whether the plugin is working for you or not, Pingdom is a great way to test whether the difference is noticeable, but it's not going to really test it to the extent that you'd probably like. I would suggest running a web page speed test on the website. You can run up to 10 page tests at a time, across quite a few countries, using variable speeds.
I'd suggest testing it in a City/State that your data center does not reside in (i.e. if your hosts DC is in Dallas, TX then run the test in Chicago, IL or Dulles, VA). This way you get a more accurate speed test from other parts of the US, or World, instead of possibly inside the same DC as your using.
You may also want to see if your web hosting provider would install an Op-Code Caching tool, such as APC, as chances are, you're going to notice more of a difference using APC than you do with Disk based caching. Once APC is enabled, you'd be able to choose that from the same drop-down as Disk & Disk Enhanced.
As for the time-stamp, using APC, until the cache is flushed, the time stamp for us doesn't change. The same seems to apply for Disk Enhanced.