Anesthesia. Quite simply, cold acts as an anesthetic. That means it decreases the sensitivty of the nerves in the area where it is applied.
Read below. It might be because cold water can decrease inflammation and pain. If you have a tooth ache though please see your dentist!
Usually doesn't. It doesn't for most people. Cold is often what sets a toothache off or worsens one. In the minority of cases where it does help, it is likely due to the fact that the cold reduces blood flow in the tooth nerve tissue and/or soothes the inflamed nerve.