Regarding the minimum operating temperature of an air-source heat pump (ASHP), it's a matter of how it was designed. All heat pumps, including ones used for heating or cooling a house, providing air conditioning for your car, or a refrigerator, use a refrigerant fluid chosen for its thermodynamic properties. Two heat exchanger surfaces are involved. In heating mode, the refrigerant must evaporate by absorbing heat at some low pressure (but above atmospheric) and corresponding temperature somewhat below that of the outside air (or ground or well water in the case of geothermal). The refrigerant vapor is compressed to a pressure at which it condenses by losing heat to the inside air (or water if that is the purpose of the heat pump); the condensation temperature must be somewhat above that of the air or water absorbing the heat. The high temperature/high pressure liquified refrigerant then passes through a throttling valve to the low side (evaporator) coil to complete the cycle. The heat rejected at the high side is the total of heat absorbed at the low side plus energy input by the compressor. The Coefficient of Performance (COP) is the ratio of heat energy delivered divided by the energy input to the compressor.
An ASHP designed for operation in a milder climate than ours might well be limited to outside air temperatures above freezing. For New England, a cold-climate heat pump can indeed operate well to -13 F or even lower. As has been mentioned, the COP drops as outside air temperature is colder. It's like pumping water up hill; it takes more pump energy to move a given amount of water higher up the hill. So with an ASHP, the greater the "lift" in temperature, the more compressor power needed to move a given amount of heat absorbed to the destination temperature. Or, given the maximum compressor input power, less heat can be absorbed and delivered over a greater temperature difference. Good ASHPs can achieve a seasonal average COP of around 3, although at very low outside temperatures this drops to 2 or lower.
By choosing the right refrigeration equipment and refrigerants, it's quite possible to absorb heat at cryogenic temperatures (think air liquifaction) and reject it to the "low" side of another complete refrigeration system, which then can reject the heat at more moderate temperatures. It's all a matter of operating conditions and refrigerant selection.
More than you wanted to know, huh?
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