Here are a few code snippets that may help you.

1. void setup() {
2.   size(screen.width, screen.height);
3.   background(255);
4.   noCursor(); // to disable mouse cursor
5. }
6.  
7. void draw() {
8.   // to force Processing's mouse position in the center of the screen
9.   mouseX = width/2;
10.   mouseY = height/2;
11. }
12.  
13. void keyPressed() {
14.   // to disable all key presses (escape etc.)
15.   // this means it will be a challenge (but not impossible)
16.   // to get out on an exported full screen application
17.   key = 0;
18. }

I guess the menu is a mac-thing, so I can't help you with that. I know for example with windows that if you kill explorer.exe the task bar disappears. Perhaps there is a similar process for mac, that runs the menu? This is more a mac than a Processing thing.

This is usually how I do it when having a computer that's just turned on/off at an event.