Arabs are definitely more sensitive to alot of different stimuli than other breeds; but they were bred to be alert to their surroundings...not dull and 'lazy' to them. That sensitivity, depending on the rider, can make for amazing rides, or terrible ones...and most Arabs are going to prefer a quiet seat, legs, hands, etc...as well as an even temper in his rider. If you are hot blooded, nervous, or easily upset, an Arab may not be the best mount for you!
Horses all have their off days. And with hotbloods sometimes their bad behavior can be more apparent and frustraiting. It just sounds like a somewhat typical hotblood spazzy moment. Haha. I've ridden many throroughbreds and some arabs and in my opinion they just tend to be more energetic occationally due to their nature, What I do with my quick horses is its just a lot of slow calm concistant riding, try trotting and walking circles, lots of gentle walk trot transitons and just a lot off just trying to cool them down a bit. A common mistake I see is people trying to work their hyper horses harder to make them calm down, in my opinion it just makes them even more worked up and hot. And for spooking just try to walk her by the object until she stops caring about it. Quick horses are the most frustraiting to deal with I think, but the most fun too :) it sounds like you did pretty well with her though. Good luck!
I just wanted to add that in my experience, yes, smaller horses/ponies are harder to stay on! In fact I've never fallen off a horse bigger than 14.3 except for while jumping (I'm horribly nervous when jumping). The creature I've fallen off the most was a 12 hh pony! Lil brat!
Anyway, sounds like part of the problem is her feeding off your nervous energy! Don't be scared of "fast", there are ways to make them stop! Learn the one-rein stop for emergencies, remember to circle her to slow her down, and I'm guessing half of the fighting was her testing you. Arabians do tend to test and get worked up if you hold them back too much. Work with your trainer and ask her advice!