Question, is the engine light on? I had a eagle talon with a similar mitsu engine as yours along with similar issues. I didn't have any overheating problems, nor did the gague seem any different. Also, the check engine light never came on. Like your car, when it was hot outside after driving or if the engine was at operating temperature it would not restart unless the engine cooled properly. I didn't have this issue so much in the winter or if I drove in the morning when it was colder, or on short trips that didn't raise the temperature of the engine much... mainly the summer was the worst, having to spend 20-40 minutes letting it cool down. The car would act like it would start but the coolant switch was making the car flood. The car had a high gas smell when attempting to restart the motor when hot due to this. Once it cooled it started perfectly, and ran perfectly fine. It was just restarting while hot that was impossible.
My mechanic found the issue in less than 5 minutes saying it was somewhat common for the coolant switch to scale over due to cooling system neglect. The mechanic asked if I used tap water to fill the coolant system, which I did, and that in turn caused the minerals from the water to crust over the switch (along with my entire cooling system). That with 9 year old coolant and it being low helped to crust the switch over as well. A quick dip of the crusted over part of the switch in a household mineral remover (not the entire switch) and a coolant system flush (with comercial coolant system flush) fixed the issue and removed a LOT of mineral deposits too.
Basically, it cost me $15 in labor plus the cost of materials to flush and fill the cooling system (and a case of beer for not screwing me over, which he could have easily).
Hope that helps..