What sort of disease is a cold-weather-related illness?

COLD-WEATHER ILLNESS. Cold-weather diseases & injuries include frostbite, chilblains, trench foot, & hypothermia. Also raynaud's phenomenon & frostnip.
Specifically. Cold weather illness can be interpreted many ways. See answers below. Specifically, cold injury is thermotrauma, affecting skin, muscle, lungs, kidneys, liver, brain, etc, as they cool down from normal operating temperature. Some reversible, some not. Sudden thermotrauma can cause shock or burns similar to those acquired from too a high a temperature.
Varies. Usually the cold weather illnesses are the "flu" viruses, but some bacterial pneumonias are more common in cold weather.