It is pretty much impossible to say where you have to improve from this description. There are a lot of different things that have to go right to make it successful.
I suggest you take a video (shot from the side opposite where your coach is spotting for the best view) ad then watch it back, in slow mo is even better. You could also watch it with your coach and ask them to point out where you need to improve. It might actually make it clearer for your coach too. It is really hard to get a good overview of how someone performs a move when you are spotting them because you are stood too close to see properly.
A couple of possibilities that spring to mind are that you may be over-arching or you may not have good enough body tension. Practising arched rocks and hold could help if that is the problem.
Imagine when you land on your hands in a bhs that the floor is lava so in order to get your hands off the ground you have to push through your shoulders. It's sort of like shrugging your shoulders. This will give you a extra push in your landing. Also, snap your feet down really quickly. You can strengthen your shoulders by doing wall handstands and holding them or handstand push-ups. Good luck!!!!!
Jump harder and use your abs!!
For the fear, just go, don't stand there and think about it bc the longer you do that, you're going to find reasons NOT to do it. So don't think about it just go.
You just really got work on your leg muscles do some calf lifts and a lot of squats you will get it quicker than you think remember repetition is the key and don't ever give up, GOODLUCK!