2 questions- 1) What is the best way to handle a horse spooking when breaking in?
2) What is a good method for teaching a horse to stay by you when you get tossed or fall?
1. Desensitizing. But not in the traditional way. I have found many things that spook my horses, from squirrels to leaves to guns. You can never desentize them to everything. You can, however, get them to control their flight instinct. Its a process but here goes (might be kinda long sorry)
Get into a round pen.
Find an object they are afraid of, but one you can take away and reintroduce easily. (I used a water bottle full of screws for the noise it gave when I shook it. My mare was absolutely terrified of water bottles when I started)
Have someone help you by holding the fearful object outside the roundpen.
Work the horse without the fearful object (by work I mean whatever they are used to doing).
While working, then instruct your helper to introduce the object (e.g. Shake the bottle). At this point the horse will let you know they are scared and revert to instinct. If they have had no training in this, they will most likely ignore you and run or spook or whatever.
Continue working the horse, by making them turn in the round pen and don't stop shaking the bottle or put away the fearful object until you see a CHANGE IN ATTITUDE. By this I mean either they stop (not likely but possible) or face you, or flick an ear towards you. Anything that indicated you have earned back their attention and they are starting to calm down. That's what you are looking for. Once they do that, cease the noise and praise them by letting them rest. Then repeat.
IMPORTANT TO NOTE: Notice I said work the horse before the stressor is introduced. That way you have his attention and then something diverts it, allowing you to bring it back. It won't work nearly as well if you just scare him right off the bat. Also notice that I said the horse needs to relax, he doesn't have to completely stop at the first try, but that's what you are aiming for. You want him to be afraid, but still be listening to YOU. That's what this will teach him.
You can do this on a lunge line, I have done it before. However, it is a lot harder since you are connected to the horse. It has its benefits and downfalls. If you choose the lung line, then after the stressor is introduced ask the horse to stop and come into you when they are afraid (keeping in mind personal space). Obvious downfall is that you could get dragged or run over. Use your judgement and keep in mind how bad the horse spooks.
2.Honestly the best method I have is the same one as above. Also, when I am thrown, I do NOT drop the reins. (I have no idea how I taught myself that but I don't). That way the horse gets a nice bump in the mouth when you fall and thus most likely won't go anywhere. If they do, obviously let go.
Also practice getting on and off with the reins in hand. It should be normal for them to stand near you.