You would have to allow the applet to script your web page; then it could access the Navigator object, which contains the information you're asking about. Of course, JS <--> Java interaction (aka LiveConnect) doesn't work in all browsers.
You can read the version of the java plug-in from an applet. Have a look at Java Tester. It has a source code listing. I don't know of a canonical list of allowed vs. prohibited properties. You'll probably just have to test the ones you want to know about.