I think you should always be ready to fall off your horse, otherwise you will always be paniking about wheather or not your going to fall off! It is literally inevitable! Its bound to happen some time!
Regards to him bucking, I would just ride it out of him! Sit deep in to your saddle for as long as you can! Sit back! And keep calm! My horse does it the odd time too when he gets a bit excited! But I just ride it off! I actually just laugh when he does it now!
I know your trainner said it would not have been the supplements your giving him, but it might just be that! Or mayby it might just be that he's just fresh!