Why does having cold hands hurt?
For question 1 it's just a biochemical response from thermoreceptor nerves in the skin, temperatures that are likely to cause lasting damage from prolonged exposure cause a neurological response - in this case it's pain.

In address to question 2 it is most likely down to the physiological response of vasoconstriction.

When the body is cold the body undergoes constriction of the blood vessels in the extremities to preserve core temperature where it is more essential for the proper functioning of vital organs. By constricting blood flow to the extremities (in this case the hands) there is less volume of blood present, because of this nerves present in the hand are closer to the surface and thus more sensitive to pain.