If the car has air-conditioning, turn on the air conditioner. The fan is set up to always run if the air conditioner is on. If it doesn't run, the fan motor, wiring, or fuse may be the problem.
Then, take note if the fan comes on when the engine is hot, without running the air conditioner. Just how hot can vary, but if it gets to the "red" zone without the fan coming on, you have a different problem, possibly the thermostat.
But yeah, it isn't necessarily a problem if its not running all the time.
Yup. You usually have two fans. The one that is directly in front of the AC compressor should only operate once the AC is turned on, and it should stay on as long as the AC is on.
The other fan will kick on when your coolant temperature hits a certain temp.