No, that results in disqualification, and is slower than going over it. Also, a common technique is kicking down the hurdles. This usually results in disqualification at the collegiate level, but you can get away with it with under-trained officials in high school.
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You don't jump a hurdle, you 'hurdle' a hurdle. Instead of jumping up then comming down, it's kind of like you take a huge step over it so you can keep running fast.
Yes you can still do hurtles. It is just that the sterotypical build for hurdles is to be lean and tall. But there is no reason why one would not be able to perform well in this event.