There is a lot of misinformation in this thread.
I can’t state definitively that NO cars use a thermostat to regulate the AC compressor, but definitely the vast majority of cars do not. Automotive AC systems are regulated via refrigerant pressure, not temperature. When you hit the AC switch, the compressor is engaged and continues to run until a pressure switch in the high-pressure side reaches it’s limit and cuts the compressor. The pressurized refrigerant leaks at a controlled rate through a valve into the “evaporator”, aka “the thing inside the dashboard that gets really cold”. It evaporates due to pressure differential as it leaks through the valve, and due to evaporation it gets really cold. The evaporated refrigerant is then passed through the “condenser” up front by the radiator, where it sheds the heat it absorbed in the evaporator and turns back into a liquid. The liquid refrigerant then goes back into the compressor for another cycle.
As far as the “temp” dial in your car goes (not the fancy-pants electronic systems where you dial in a temperature, but the standard “blue-through-red” dial), all that does is control the amount of HEAT being added to the cabin air, either by controlling the flow of hot engine coolant through the heater core, or by controlling the amount of hot air from the heater core that is allowed to blend into the cabin air. In other words, by moving the dial from blue toward red, you are adding heat from the engine to the cold air produced by your AC, and thereby “undoing” some of the work it has done to move heat outside.
So to answer the OP, your husband is wrong. In a properly working AC system, the temperature of the evaporator is always the same. You merely control the flow of air over it with the fan switch. Adjusting the blue-to-red knob has NO effect on the function of the AC system, it just turns on the heat also.
2nd question: not enough information. Manually cycling the AC system results in loss of efficiency because the entire system needs to be brought up/down to pressure/temperature every time you turn it on, and that wastes energy. It’s a question of whether the energy you save during the periods that the system is switched off exceeds the energy wasted by repeatedly cycling the system.
Just FYI, if the AC is too cold for you, it’s better from an efficiency perspective to turn the fan down rather than turn the heat up. Turning the fan down does reduce the cooling load on the AC, turning up the heat does not.
Further FYI- not all car AC compressors cycle on and off. Newer VWs (and some others) have variable-output compressors that dont’ need to cycle all the way off.
All this is further confused by the fact that some (usually domestic POS) cars are designed to run the AC whenever the “defog” setting is selected, presumably because the designers thought the average consumer isn’t smart enough to manually use the AC to dehumidify as well as cool the air.