User action is required; the applet needs permission to get outside of its sandbox, so a standard "This applet wants permission to access your computer and data, Approve or Deny?" dialogue is presented.

Drive by download & execute vulnerabilities exist in Java for many browsers.

So yes, it is possible for the "trojan" to work as described.

If "as described" includes running automatically (without user permission), then you'd be incorrect in this case. This particular trojan doesn't include any privilege escalation exploits, so the user must confirm a couple of security dialog boxes (including keying their password) before the trojan can install.

This is not entirely true. There will be only 1 popup asking the user's permission to "run" java code through the browser. After that, the applet can download anything on the box it needs and execute it.

The applet has full access to the local filesystem with the same priviliges the original user has. If needed the hackers can further exploit the machine, by escalating user priviliges with some corrupt scripting..

One click is enough to seriously damage your machine, be careful;) This is what the popup looks like in Firefox http://www.ussu.ca/studentgroups/JavaApplet.jpg