This is a common problem on this car. It is caused by a defective coolant temperature sensor. When the sensor is acting up, the gauge gets a cold reading and will drop to cold.
When the engine is cold , the oil pressure will be high due to the oil being thicker. Because the car is actually hot, the oil pressure is lower as it should be, but since the coolant sensor is sending a cold reading the instrument cluster is expecting a higher oil pressure. So you get a false oil pressure light.
Hope that makes sense.
Yes, a new coolant sensor should correct problem.
The check engine light will come on if the ECM sees an open circuit or a short circuit inXXXXXcycles.
But there are 2 separate sensors in the sensor housing. There are 4 pins, 2 are sensor for the ECM. The other 2 are sensor for the instrument cluster. Often 1 side fails and the other doesn't. If you have access to a scan tool that can faults in the instrument cluster, you would likely see a fault in there.