Pathfinder isn't a system I have explored in depth, but I do train horses, so I can cast some light from that side of things. Horses have very strong, powerful jaws and even a relatively mild bite hurts a lot. A severe bite can certainly take a chunk out of you or sever fingers and a horse that habitually bites people is dangerous to be around.
If you really want to use something else that makes sense in terms of how a horse fights, they will kick out with their back feet ( with bone-breaking power ) if something they are concerned about is behind them, but a trained warhorse would be more likely to strike out with a front hoof. This is an unbelievably fast forward kick with the front legs, which you would usually only see in nature from an aggressive stallion. They don't rear up, they simply lash out. If someone is knocked down ( and a horse can knock down most people just by running into them ) an aggressive horse might paw at them on the ground. The capacity for aggression is closer to the surface on stallions, which is one reason they tended to be used as warhorses historically.
These are natural behaviours, but by nature a horse is way more flight than fight, so if you are concerned about realism you would need a well-trained warhorse to use them in battle. The average riding horse would be more likely to be intimidated by that kind of situation and be inclined to try and get itself ( and possibly you ) to somewhere safer, as quickly as possible.