Hey gymgurl, I watched your vid the other day but didn't have time to post, just watched it again and these are my tips:
In the gym:
1) Get yourself a large safety mat, the 30cm deep ones are good. Stand with your back to it, so your ankles are actually touching the mat. Fall backwards with a straight body. As you land, your feet should lift off the floor, but not higher than the mat. In other words, you should be falling into a straight shape, not dished. Do this a few times until you are happy with the feeling. Really try to engage your muscles as you fall - particularly in your stomach, buttocks and legs. It wouldn't hurt to point your toes either!
2) As above but add a little push backwards. I'd do this with the arms by the ears at this stage. Just stand up straight, bend the knees and upon straightening them, push yourself backwards to land flat on the mat.
3) Add in some shapes and arms. Start by standing with a dished body, arms by ears. Ensure your bottom is tucked in and stomach pulled in (pelvic tilt) and that your knees are locked tight. Your start position isn't bad just work on that pelvic tilt and locking your knees:
Begin to fall backwards (imagine a piece of string is attached to your belly button, and comes out through your back, gross I know, but still! Then imagine someone is pulling you backwards) As you begin to lean swing your arms down and back. At this point your knees should be in front of the rest of your body.
As you can see your shoulders are too far forwards at this point. Where your shoulders should be:
Now jump backwards onto your back again. Really think about extending your knees. Push them into the floor andreally try and launch yourself backwards by extending your knees.
4) Now remove the mat, but add in a spotter. Do your bhs with a spot, over and over again. Your coach will be able to stop you mid bhs and correct any shapes. I notice on your video that at around 5 seconds, you start to move your feet.
This completely eradicates any 'lean' that you may have done previously as your shoulders will be even further forwards than they should be. With a spot you will be able to gain confidence to lean and jump without shuffling your feet. With a spot you will also learn the correct 'arm swing' timings. I usually say to my gymnasts, Stand, Lean/Fall, Begin to swing arms down, as arms get level with hips, jump backwards and swing arms quickly upwards.
As you jump back into the bhs think about making a long rainbow shape with your body. This position isn't bad, however had you pushed your feet into the ground more on take off your bhs would have been longer, thus making it easier to keep legs straight.
Unfortunately what happens in your bhs is that your hands come down pretty close to where your feet have just been I've drawn a red line on this one to show roughly what your body position should be
As for the 'snap' from hands to feet, I'd work on donkey kicks over and over again, with straight legs. Get your coach to spot you and correct shapes.
Your bhs really isn't that bad, but there are lots of little things that can be fixed to make the whole thing look and feel better, particularly really learning the timing and to push your legs straight. Good luck - If this post has been helpful I'll do a similar one for the second part if you'd like.
I think it is great that you care about improving your form as I've said in another of your posts, so I hope this post has been encouraging rather than 'picky'!