[callme]

Сам stage не реагирует на MOUSE_OVER. Его дети в дисплей-листе реагируют.

Вот немного переписал код. Курсор сразу скрывается при запуске приложения как вы хотите. И продолжает нормально работать после фуллскрина.

package
{
    import flash.display.Bitmap;
    import flash.display.BitmapData;
    import flash.display.Sprite;
    import flash.display.StageDisplayState;
    import flash.display.StageScaleMode;
    import flash.events.Event;
    import flash.events.MouseEvent;
 
    public class Main extends Sprite
    {
        public function Main()
        {
            var bitmap:Bitmap = new Bitmap();
            bitmap.bitmapData = new BitmapData(800, 600, false, 0);
 
            var sprite:Sprite = new Sprite();
            addChild(sprite);
            sprite.addChild(bitmap);
 
            var mainMenu:MainMenu = new MainMenu();
            addChild(mainMenu);
 
            stage.addEventListener(MouseEvent.CLICK, stage_click);
        }
 
        private function stage_click(e:MouseEvent):void 
        {
            stage.scaleMode = StageScaleMode.EXACT_FIT;
            stage.displayState = StageDisplayState.FULL_SCREEN; 
        }
    }
}

package
{
    import flash.display.Bitmap;
    import flash.display.BitmapData;
    import flash.events.Event;
    import flash.events.MouseEvent;
    import flash.events.TimerEvent;
    import flash.ui.Mouse;
    import flash.utils.Timer;
 
    public class MainMenu extends Bitmap
    {
        private var _timerIdle:Timer;
 
        public function MainMenu()
        {
            bitmapData = new BitmapData(100, 100, false, 0xff0000);
 
            visible = false;
            Mouse.hide();
 
            if (stage) initMonitoringMouseMove();
            else addEventListener(Event.ADDED_TO_STAGE, initMonitoringMouseMove);
        }
 
        private function initMonitoringMouseMove(e:Event = null):void
        {
            removeEventListener(Event.ADDED_TO_STAGE, initMonitoringMouseMove);
            stage.addEventListener(Event.MOUSE_LEAVE, onMouseLeaveHandler);
            stage.addEventListener(MouseEvent.MOUSE_OVER, onMouseOverHandler);
 
            _timerIdle = new Timer(2000);
            _timerIdle.addEventListener(TimerEvent.TIMER, onTimerIdleCompleteHandler);
 
            onMouseOverHandler();
        }
 
        private function onMouseOverHandler(e:MouseEvent = null):void
        {
            stage.removeEventListener(MouseEvent.MOUSE_OVER, onMouseOverHandler);
            stage.addEventListener(MouseEvent.MOUSE_MOVE, onMouseMoveHandler);
        }
 
        private function onMouseLeaveHandler(e:Event):void
        {
            trace('mouseLeave');
            stage.removeEventListener(MouseEvent.MOUSE_MOVE, onMouseMoveHandler);
            stage.addEventListener(MouseEvent.MOUSE_OVER, onMouseOverHandler);
        }
 
        private function onMouseMoveHandler(e:MouseEvent):void
        {
            visible = true;
            Mouse.show();
            _timerIdle.delay = 2000;
            _timerIdle.start();
        }
 
        private function onTimerIdleCompleteHandler(e:TimerEvent):void
        {
            _timerIdle.stop();
            visible = false;
            Mouse.hide();
        }
 
        public function destroy():void
        {
            removeEventListener(Event.ADDED_TO_STAGE, initMonitoringMouseMove);
 
            if (stage)
            {
                stage.removeEventListener(MouseEvent.MOUSE_MOVE, onMouseMoveHandler);
                stage.removeEventListener(MouseEvent.MOUSE_OVER, onMouseOverHandler);
                stage.removeEventListener(Event.MOUSE_LEAVE, onMouseLeaveHandler);
            }
 
            if (_timerIdle)
            {
                _timerIdle.stop();
                _timerIdle.removeEventListener(TimerEvent.TIMER, onTimerIdleCompleteHandler);
            }
        }
    }
}