I'm no expert in zombie anatomy but it seems that without blood flow, movement would be impossible as muscles and tendons would solidify. It is my opinion that the body is reanimated after death leaving primal instinct driven (possibly due to neural damage from an oxygen starved brain) semi-rigid corpses. This being said, the level of mobility may be directly related to the time that the body spent "dead".
As for bleeding from a gunshot wound, that also depends on the time after death. Blood will stay red and liquid within the body for many hours after death. In the case of 'runners' the body is still animated however, brain function has been so impaired as to cause the subject to act in a primal manner.
Tests are required to determine the exact answer to your question. I hope my assumptions help your pondering.