Try to keep your hips square before do the back handspring and look at your hands when you are in the back handspring. I am training a back handspring right now and that is what my coach told me.
On your back walkover, just try to picture your coach's hand always there. That's what I did and it helped me a lot! Also, try to think of "buzz words" that help you throw it like repeating - the coach is right there. It really helped me. And on the rod floor, if you tumble in a straight line...