1999 Other Jeep Models Coolant Tem Gauge and Battery

I took my car into a local garage and needed the battery cable and connector replaced due to short and poor battery operation. After driving 5 minutes the Coolant Temp. Gauge went to the danger zone. I look under the hood and the battery was extremely hot. The next morning. I checked the radiator fluid and and reservoir and it was all okay. When at a light or the car is not in drive the gauge is in the middle. If I drive over 45 mph it creeps back into the danger zone.

The thermostat is not the issue because it has been replaced yet the issue continues.

I took the vehicle back to the garage and they are still working on it 24 hrs later. Could you help me help them.

The follow-up is that it was determined to be the water pump. Could somepne please explain how the water pump ineffectiveness created the high gauge temp and the battery to be hot?