Heat pumps do not usually run air conditioning functions in cold weather. However, if you were to set your heat pump to cool your house during winter, it would be able to do so easily. Heat pumps cool by gathering heat from the air in your house and moving it, via refrigerant, to the air outside. The cold conditions actually allow the pump to release heat more easily.
The cold weather can damage heat pumps, but that does not depend on what functions the heat pump is performing. Ice can build up on the outdoor coils. With the ice blocking the heat transfer, thermal energy cannot pass as easily and the heat pump will not operate as efficiently. This ice can also wear down and damage coils.
When a heat pump tries to heat in cold weather, it runs into a problem. Moving heat into cold air is easy, but drawing heat from cold air can be very difficult. If the temperature drops below 45 degrees or so, the heat pump may not be able to create much heat at all. This leads to long running cycles and the eventual failure of the heat pump, but most pumps have safety features that shut the pump down when it cannot heat correctly.
The cold weather weaknesses of heat pumps may make it seem unlikely that they would be used in colder climates. However, many pumps have an electrical heating element installed that can create heat in the event of cold conditions. The electrical element is much more costly to use, but it can help protect a heat pump when necessary.