"Left and right" is an informal expression meaning "everywhere." It's usually used to imply that there is too much of something, or that something is violent. No, you cannot use it to refer to parts of your own body.

As Phil said, it means something is happening frequently, excessively, carelessly, or in many places. You wouldn't normally say that you're "hurrying left and right" (maybe because it sounds too literal, as if you are really hurrying to the left, and then to the right). You could say, "I've been answering phonecalls left and right," or "I've been dealing with problems left and right." It's usually used for describing something negative. (My Chinese is not very good, so I'm not sure, but based on examples I've seen, I think the meaning may be similar to 動輒 or 動不動.)