Is there any way to turn off screen and adjust volume on the iPad programmatically? It can be private API as this is for personal use. I know that in iPhone you can use proximity sensor but I don't suppose this method would work on iPad.
You can use proximityMonitoringEnabled and proximityState to handle screen. Look at comment in this answer.
And there is no way to adjust volume
From developer.apple
Global system volume, including your application's volume, is handled by iPhone OS and is not accessible by applications.