When I deploy a new app or quit it over control panel, what is done behind the scenes?
Do you kill the app?
or do you send a signal or trigger the app to call Quit, so destructors for various objects like database connections can run?
We look up the process ID and kill the app. The Quit mechanism in the control panel should really be used as a way to quit an app that is not responding for some reason. I suggest that you add a call to App.Quit in your application somewhere if you want to cleanly quit your app.
If you would like signal handling in console apps, please file a feature request.
I have signal handling in web apps via plugin. 
Then look for SIGTERM.