As Stephan says: maybe you can use a mouse motion listener and open the menu when the mouse enters the component. Careful, though. If you have too many drop down menus appearing as you move the mouse around, the display will be very difficult to use.
Add a MouseListener to a JMenu that invokes the JMenu.doClick() method on a mouseEntered() event.
You can write web app GUIs in Java using GWT, but personally I'm not a big fan of it. I prefer to use HTML and CSS directly, and use an MVC web framework that supports templating.
Java is not run in browsers directly (anymore). There are client-side languages such as ActionScript, Flash, Silverlight, etc., but JavaScript is the most common.
For web applications? I'm a firm believer of MVC frameworks, and Java has a couple of really good ones, such as Spring MVC. It still means that you will need to know at least CSS and HTML, and JavaScript is always good to know.
The frameworks run server-side. The only thing that 'runs' client-side would be JavaScript, and you can do a lot of fancy effects with just CSS instead.
Yes, and Applets are old technology that should not be used anymore. Java is used server-side, usually with frameworks, to serve web pages, that may or may not be enhanced with JavaScript on the client-side.
You can make an online game with videos completely in Java on the server-side, without resorting to browser plugins. All you need is Java with a web framework, CSS3 and HTML5.
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I don't know. I've only used Jersey and Spring so far.
Don't get me started about those stupid light bulbs.