Usually you start off by doing VERY small jumps, practically just trotting over poles, so yeah it can be scary to move up to big jumps but you normally get used to starting small first.
It feels a bit like being on a rollercoaster, your stomach sort of flips as you go over the jump!
As for staying in the saddle, I've never ridden Western so I assume you guys get to hold onto the horn if you feel the need? Well, with English you're really building up muscles you never knew you had, using your butt and legs to stay attached. Using your thighs, calves, butt and keeping some weight in your heels will help you stay in the saddle. It takes a lot of work and I don't recommend anyone try to jump straight away, you really have to work up to it!
Hope that helps :)
Satin Reign aka "Misty"... my life, my love, my everything.