First of all, what it sounds like is you've got a heat pump. A heat pump is an air conditioner that has a three way valve allowing for it to reverse the operation - so that it can function as a heater or an air conditioner without using electric heat strips, or a gas furnace.
It will have either electric heat strips or a gas furnace paired with it, but those will not get employed unless the temperature drops enough that it can't keep up with the demand for heat inside the home without their use.
So when the three way valve, also knows as a reversing valve gets stuck - the unit deposits refrigerated air outside in air conditioning mode, and hot air inside.
Did that make sense? You're question describes a heat pump central heating and air unit with a stuck reversing valve.
Now, the thing about your fan motor is another problem entirely. Yes, a run capacitor MIGHT BE needed to fix that problem. But it could also be that your fan motor's bearings are shot, and if that is the case, then the motor itself should be replaced. The person replacing it should always replace the run capacitor along with the motor.