Cooling fan runs intermittently

It's normal for the fan to switch on and off. It is controlled by the computer and a couple sensors. When you are driving down the road the airflow is normally enough to keep the car cool and the fan stays off. When you get into town and or stop and go traffic the fan will switch on and off and switch speeds as needed. The fan working with the AC on is also normal, it pulls cooling air over the condenser to help the system cool better when you are driving slowly. As to the AC not working unless the fan works, it's actually the other way around. From the sounds of it you could simply have a low refrigerant charge which would cause issues with it working. I would probably put a set of AC gauges on it and see if it's low first. For that you can get the gauge set as a loaner tool at many parts stores. Read the end section of this guide where it talks about the pressures and what you should see on a charged system.
https://www.2carpros.com/articles/re-charge-an-air-conditioner-system

As for taking it to a dealer, most shops could also work on it and probably for a lot less than a dealers shop.

They both work the same way. When you turn on the AC it activates the fan to draw air across the condenser. However the AC should work even if the fan doesn't come on. That is because once you start driving the fan shuts off as you hit a speed where the airflow is high enough to keep the condenser working.
They are interconnected that way, pretty much all of the modern vehicles with electric fans work the same way. The system is turned on by either a switch or relay, the ECM looks at the ref. Pressure sensor or switch. If the pressure indicates there is enough of a charge in the system it commands the clutch on. At the same time it sends a signal to the fan control relay(s) to turn the fan on. It then monitors the pressure sensor and various others to keep the refrigerant in the correct operating pressure zone. If you have the automatic HVAC where you set a temperature and the system does the rest is is a bit more complicated as those have a lot more sensors to watch.