Hi, I have a similar problem and live in Santa Barbara, where it is not cold. I hear the solenoid click and all electric functions work. I've noticed that if I twist the key different, either a little more pressure in or down I can sometimes get it to work. Sometimes the engine will start to kick over, but then dies in about 1/2 to 1 second. When it does start, it runs great. It does not always happen. I may go a week with no issues, so it is not the car battery.
Davis