fan is kick in when ac turn on or engine reach the high temperature to be cold down by order com from the engine temperature sensor
to the ECU THEN TO THE fan RELAY .some car fan have own ecu . if the car not overheat and the fan kick in then shut of kick in shot of this is normal if the car overheat LOOK for air bubble in the cooling system bad thermostat , faulty temperature sensor . relay . ECU .
yes the engine have to be running in order to have heat .the blower motor that plow the air have resistor to control speed if the resistor faulty it may not work in some speed or kick in at all or some time it work some time it wont