2000 Mercury Villager power steering

It sound like you have two issues here. Check both fluids with the engine off. Is the coolant full? Check both the reservoir and in the radiator cap when engine is cold because opening the radiator when it's hot could cause serious burns.
Check the power steering fluid. Is there any leaks? Some times the dust boots on both ends of the steering gear will start to fill up with fluid if your not able to see any leaks. Any air in the power steering (ie leaks) will cause loss of power assist in the steering and cause growling/whine noises.

For over heating air in the cooling system will cause overheating so make sure it is completely full in the radiator and the reservoir. A missing belt no longer spinning the water pump would cause overheating. These water pumps have a history of breaking they use plastic fins to pump water which eventually break no longer pumping coolant through.

I don't think the overheating or hard steering are connected (besides the serpentine belt, the belt is the only connection). I think you have two issues here. For hard steering the fluid level has to be full and that there is no air within the system. If everything is good there then I would suspect the power steering pump. Inspect all the power steering lines for being pinched or anything causing a restriction. If no restriction found most likely your power steering pump has failed, the next possibility is the steering gear.

It charges about 2-2.5 hours to perform. You'll need to remove the pump from underneath the vehicle and it requires some special tools to remove the pulley and transfer it to the new pump and bleeding the air out the power steering system.