thermostat controls temperaturte of the engine, it opens to send coolant on big loop, closes to keep coolangt on small loop - if it is stuck in one position, the best is when it is stuck in open position, when the coolant goes on big loop, thus the engine is not overheating but rather stays below operating temperature, which is bad too, just not as catastrophic... the thermostat does not measure temperature, it only opens and closes based on the measurements taken by the temperature sensors... there is one sensor unit with two sensors built-in, one sensor actually sends the info to the engine and the other sensor sends the info to the cluster... in absense of the correct sensor readings, the computer substitutes the temperature value based on other temperature readings (air, oil, etc) but it is not a good option, as the performance is reduced/compromized... you might not feel difference in newly found performance but you might only lose 1-3 horsepower, so to say...

as far as the check engine light, best to read the codes and see what it says, as the light comes on for various reasons...