Air Conditioner - Outside vs Inside Temperature?

The air conditioner will cycle according to the thermostat setting to maintain the set indoor temperature. It won't run much in the cooling cycle if it's cool outside.

If it's extremely cold outside, there's a risk of refrigerant condensing inside the compressor, which can cause damage. Central air conditioner compressors, and those used with heat pumps, contain a crankcase heater to prevent this. A cheap window air conditioner may not have a crankcase heater.

The AC kicks in when the inside temp is above the set point and moves heat outside.

The only difference when the outside temp is below the inside temp is that the inside temp probably won't go back up.

Depends on how low. Houses/people generate heat, insulation/buildings release heat or gain heat to the outside depending on insulation and the difference in temp. All that means is that in some cases the equipment and people can generate more heat then the rate it escapes. So ones air conditioner can kick on and run in that situation. If the air conditioner is projected to run in cooler temps on a regular basis then things like fan controls are put in place, that operate according to the internal pressures of the system. Of course the expected operating temps dictate what one should do, but for most residential units, it isn't even a concern. In say compute rooms in the north where there is a need to run air conditioning in the winter, in freezing temps and such. Not controlling those internal pressures of the system will result in damage to the system, and then damage to that equipment

The air conditioner will do what the thermostat calls for. Inside, you are generating heat, your appliances generate heat. The thermostat measures the temperature of the space it's in, not the outside air.