Must I reward my child with money for doing chores?

Trade works. Parents forget they automatically do things their kid values which can stop. That trip to practice/club/etc. Can be traded for work they do. You don't have to sign off on that field trip at school, they can cool their heals in the library unless they will trade it for chores. Get creative. While money can work in some instances, teaching them to put value on effort goes both ways.
No. There is no reason that a reward for any type of behavior/task/accomplishment needs to be monetary.