Going backwards through your questions.
Why is it that MOUSE_MOVED in MouseEvent is considered an int?
All of the mouse constants are ints. That's how enumeration was done in Java before version 1.5.
How do I find the time since the mouse last moved?
In your mouseMoved method, you save the current time somewhere in your GUI model. You then write a method in your GUI model that returns the idle time.
idleTime = System.currentTimeMillis() - savedTimeinMillis.
You divide the idle time by 1000 for seconds.