Why is my shower faucet turning itself off?
It sounds to me like your diverter valve (the round piece w/ the green on it in your pictures) is failing.  It may be as simple a fix as pulling the valve and replacing the gaskets (O rings, probably) and reinserting the valve, or alternatively just replacing the diverter valve entirely.

Unscrew the chrome cap surrounding the diverter valve and the valve itself should pull right out. Take that to a plumbing supply shop/lowes/home depot/someone who carries your brand of faucet.