The fan should come on low speed when the A/C is on. It is also triggered by the temperature switch on the right-hand side of the radiator which has two temperature set points, one to initiate low speed and one for high speed.
The most common failure is that the contacts on the low-speed side of the relay burn up after a decade or so of use. If you unplug the temperature sensor and short across it to activate the fan and you find that you have high speed but not low speed, this is the most likely problem.
Another possible failure is the temperature switch. Not common, but anything can fail on a 13 year-old car.
The fan uses a resistor at the fan to get low speed, and that can fail as well. Fans are expensive, and I don't think the resistor is sold separately, so if my resistor failed, I would try to wire around it so I had high speed fan in all situations.
You mention climate control problems. The common problem in your climate control is cheap and easy to fix.