This is all normal. As the car is shut off, the water quits flowing, or being cooled. The engine temperature which was being controlled by the water is no longer being lowered. It sits and bakes. The poor running you describe could be related to your fuel heating up in the rail, beginning to boil. After your starts back up, fresh coolant replaces the hot coolant in your block, bringing the temperature back to more normal levels. Also, once you circulate fresh fuel into your rail, your car will run much better. Common problem, that. Curious if you have the fuel rail cooling system or not. It is designed to cut on around 230 degrees or higher (don't know the exact temp, but it is high) and blow air on your rail and injectors. If so equipped, you might investigate and see if your is working properly. Do you hear the fan when you come out of the Quickie-Mart? This fan works with the key off. May be a bad sensor (look at your thermostat sensor area), or a bad connection to the sensor. Possibly your fan motor has failed.
Look up heat soak in the archives for possible causes and solutions.