1) It depends on the Android version.
2) This has been discussed many times here. Search Kiosk for example.
3) What does MX Player is very basic. It closes the status bar when you try to open it.
Code here. If you're fast enough, you can use the Home or Back key in lock mode.
It is possible to redirect the Home key to your app on an unrooted phone by making it the launcher app, but personaly I won't explain how to do. If your app crashes, that may brick your device (I had to reinstall Android more than once during my tests).