I'm not suggesting doing this or whatever, I just want to know the answer. Let's say it is 80 deg F outside and I put the AC on 50 deg F. Obviously this will cause the AC to run all day, would this running all day cause the motor to burn or would the AC be fine? Would the Compressor lines freeze? Again this is a general, model agnostic, question. You can assume it's a good functioning unit with no obvious problems.
With a properly functioning system no the motor will not burn up. And the coils and line set may show some frost but would not freeze up.
If you leave the system running and there is a drastic temp drop things might ice up a little but the throttle valve will start restricting the flow of refrigerant so the compressor may start cycling , systems usually have a minimum off time so the cycling will not over heat the motor. For systems with simple orifices the pressure switch turns off the compressor until time to start again. Most all residential compressors have internal thermal overloads that stop the motor from operating if they get two hot. This is yet another safety so no a properly operating hvac system will not burn up if left on all the time.
Compressor motors are self-cooling and designed to work for extended periods. Obviously more runtime means more wear, but there's no point at which it'll typically overheat or suffer other damage.
Refrigerant lines shouldn't freeze. In theory enough heat is taken out in the exchanger that it never gets out of operating range. Refrigerant stays liquid far below where it should ever be in use--around -150°F.
What can happen, though, is ice buildup. If you actually cool the building to such low temps, relative humidity rises dramatically. This leads to condensation and ice. Eventually the heat exchanger becomes completely clogged and the system breaks down. Then it runs without actually doing much, just wasting energy. If the weather is warm outside you'd be hard-pressed to get that to happen, though.
It shouldn't. When our previous house was for sale an incompetent house inspector set the thermostat to about 55 F. We were not home and it ran about 36 hours. The outside temp was only in the low 70's so the thermostat had to be turned low to run . The house was cold when we got home but otherwise no problem. The incompetent part was about a water heater TP valve.
This often occurs when an a/c system is undersized. It will run continuously and never cycle off because the system does not have the cooling capacity to reach a reasonable set point like 75 F. This does not cause any problems except I think it can cause excessively dry air inside.
We have a 3.5 ton unit in a 2000 sq ft poorly insulated house. On a very hot day with the set point at 75 F, the unit will come on at 11 am and not cycle off till 4 pm or later. The unit has been in service for 29 years (orig compressor and air handler motor).
The explanation of excessive drying of indoor air with an undersized a/c is that when the evaporator is cycled on it still removes nearly all the moisture from the air passing over it. The claim is that an evaporator with more cooling capacity does not remove significantly more moisture than one with less cooling capacity while it is cycled on. And since a properly sized evaporator will be cycled off a lot of the time, then the relative humidity in the house will be higher with a properly sized evaporator. Some fancier units can control both temp and humidity. I think they do this by regulating the cooling capacity of the evaporator which affects the relative duty cycle.