Why does my wrist pop and why do I feel shooting pain from time to time?

Depends... Altered wrist mechanics from a cartilage or ligament tear can cause abnormal, and normally painful clicking around the wrist after trauma. Some nonpainful clicking around the wrist can be normal as long as they do not impede motion or cause discomfort. Given your pain you likely need xrays and a good evaluation by a trained professional.
A popping wrist. Maybe caused by a variety of issues or it may be just a simple natural normal occurrence often when one has pain in list start looking at a certain way to show that pain you start to notice little pops that may have been there. However popping may also mean that there is a ligament injury or a swollen joint and any proper evaluation by a hand surgeon can be great help.