I come from a slightly different perspective... I'm a trainer who uses school horses but doesn't ride them! (unless they need a tune up) I do however know them very very well!
Do you own a horse? Yes
How do they compare to a riding school horse? He is more sensitive to aids, has a higher level of training, and more fit with his body (muscles developed properly)
Have you ever been bitten/kicked/thrown off a riding school horse? Not in a long long long time. As far as the students go... very few have been bitten (I hate biters, refuse to by them for the school) and usually involve a treat and a wayward finger. No one has been kicked (again, I don't buy kickers). Thrown off? Well, yes. That does occasionally happen. :) More frequently from certain horses rather then others. More likely when someone falls off it's because they've lost their balance (riding without stirrups, learning how to canter, etc) or got to a bad jump. There is a pony who tends to spook and peek at fences. I only put advanced kids on him but every so often he'll get one of them.
When riding a school horse, do they seem to enjoy being ridden? This is a difficult question. Lots horses are prone to trying to get out of work (ring sour). We have one or two that are very girthy. But I don't think this necessarily means they hate their job. Only a handful can't wait to be ridden. But this I've noticed.... when it comes to the older guys who have worked their entire lives, retiring them early often makes them go downhill very quickly. We have several horses that are in their upper 20's and even lower 30's that do little w/t lessons a few times a week that look fabulous for their age. While some of them may not seem as enthusiastic about their jobs, I firmly believe having a job benefits almost all horses (soundness problems aside) and they enjoy having a purpose to their lives. When a horse hates their job, they let us know and they are simply not good school horses. That's when we find another home for them.
When not being ridden, do the horses seem more relaxed? Not really.