I agree with the people above about working on your snap-down to help with pulling your chest up. Having a good snap-down is largely dependent on your core (abs) and leg strength. Try some conditioning to help with building up strength - I'd recommend '300 Abs' for your core and Whitney Loves 'tuck' conditioning for your legs (you can google both).
Your coach is also right about pushing off your shoulders more. What they mean is that when you hit the handstand position in your backhandspring, you need to 'pop' with your shoulders, as if you are shrugging. This'll help with snapping down fast from the handstand position to a dish position/standing up.
With your elbows bending and your head touching the floor, this sounds to me like you are not 'locking out' your arms enough. Locking out is when you squeeze all of your muscles so that they are very taut. Try standing with your arms above your head with your palms spread out, and squeeze them. Get someone to push down on your palms - if your arms don't bend, they're locked. That's what you need to try and do in your back handspring.