This may not be required, but there are no system properties that identify a NetX-launched program as either NetX or Web Start. Sun's implementation sets javawebstart.version (and probably a couple of other deployment properties).
In AWT mode, the current component under the mouse is used as the target when performing multiple clicks.
As for the second condition, sharing the UI when doing UI tests is always problematic as long as the Java event queue is still receiving events from the system. Consider using a dedicated system for your tests or setting up a virtual user desktop.
It's not generally possible to determine the character for a given key code; whether that field has a value other than CHAR_UNDEFINED is platform and VM-specific.
You need to provide the appropriate values in the AppletContext passed to your applet on initialization. Make getDocumentBase() return something non-null.