Not possible with LWJGL (native) Display. You can't request focus. A future release of LWJGL (3.0) has some things like this planned, but until then you might be able to achieve this with CanvasGameContainer + JWindow parent.setVisible
Not possible with native display. You could achieve this with CanvasGameContainer.setOpacity
Not possible without diving into native code and implementing it yourself for each platform. You could try CanvasGameContainer with a fancy JFrame, e.g. some Java 7 code.undecorated
You can get this with a hidden LWJGL VM switch.
Reliable? I'm not sure. Maybe JWindow or an undecorated JFrame + CanvasGameContainer would be more reliable. I'd say test it out and see what works best for your game and your target audience.
If you want to add things like transparent windows, focusing, etc. to the LWJGL display, you'll need to poke around with the native code and implement it differently for each platform that you're supporting. Possible, but not easy.