Q: Cooling temperature sensor light on.

Coolant temperature sensors generally will not record fault codes in the engine computer or turn on the Check Engine light unless they completely fail by either a short or open circuit. When the coolant temperature sensor fails completely you will either have some drivability problems such as reduced power of lower fuel economy (open circuit) or the cooling fan will run continuously even when the engine is cold (short circuit). In my experience what often happen is that the coolant temperature sensor will still read within its range but will not read correctly as it ages. This will not record any faults but will cause the temperature gauge to read incorrectly. Now the first thing to check is to see if the coolant temperature sensor is reading correctly and that the cooling system is operating correctly (ie. the thermostat is operating correctly). You may want to enlist the help of a mechanic, such as one from YourMechanic, to run a performance check on the cooling system and compare actual coolant temperature versus actual coolant temperature. This will narrow down the cause of the temperature reading.