Air Conditioners? ?

People are supposed to turn on their heat in the winter because it will make your house feel warmer. You're doing the opposite which will result in colder air but 20 degrees is a stretch.

No, it wouldn't work if the outside air was colder than inside. Also, it can't drop the temperature any lower than the refrigerant fluid itself, which is about 40-50°.

You would not be able to get an air conditioner to make the house that cold.  Because of the way an air unit works, if the temperature outside is too cold, the air unit doesn't cycle correctly and will not get colder.

You would get a colder home by just turning off the heat and opening the windows instead of trying to run the air conditioning.

One reason I know this is because I have tried to use my air conditioner in the winter when my home got to hot while cooking.  It is a great unit, but it barely knocked the heat off from the cooking.  It is because it can't correctly cycle when the outside air is too cold.  

When I was younger I had a box truck and I took a $15,000 BTU air conditioner I jury rig thermostat.

I was able to get the temperature in the truck down to 35°