If the water sensor is 'Bad' the ECU will approximate what the water temp could be.. and it will be very predictable, and it will 'Start Cold' every time you restart the Engine.. You can check it my looking at the ECU's water temp from the Laptop.. let the temp rise, then Shut it off, and restart.. if the sensor Info is 'Synthetic' it will start displaying the water temp as cold again.. This still does not explain the entire scenario you are seeing, but this is all I got presently..