Warning lights

After driving across town yesterday we made a stop for gas. When I tried to restart the engine the battery was acting as if it were dead. The lights were dim, the engine would only click, and the clock display had gone back to 12:00. After getting jumped started, we drove to Batteries Plus to purchase a new battery (5 miles). I had my wife shut the engine off and open up the hood. Then I realized the battery was not easily accessible. I then had her restart the engine. It did not want to start. She played with the throttle (pedal) a little bit and managed to get it started. Then the lights came on. The engine light came on and there was a message on the odometer screen. This message read "Engine Hot, A/C Off". We also noticed that the thermostat (needle) was only registering "C". We delicately drove it home, stopping every now and then to verify the engine wasn't getting too warm. Once we got it home, I checked under the hood and all looked and felt good. No real hot spots, no sweet odors, and no bad hoses. At no time during the trip home did we have any issues with the engine. No power loss or weird noises. The engine responded as normal. My question is did a fuse or sensor possibly fail? Or is it possible something happened when we jumped it? The vehicle we jumped it with was of a similar type, with the same voltage of battery. I checked the voltage of our battery when we got home. The battery was holding 12.64 VDC. I will be replacing the battery, but I'm nervous about the warnings. Thank you for any insight you can give.