yes this is normal for the AC to cycle even when the AC is off. Do not be concerned with the fuel consumption or it tearing up the pump. It does this to keep everything lubricated . if it did not do this seals will dry up and it will leak small amounts of Freon, it also lubricates the AC compressor and if you have the option to turn the AC on with the heat on this will remove the moisture in the car while keeping you toasty

The A/C system is both operating with A/C on and Heater as it assists in keeping the heated air from being too dry. If it runs with the fan inside turned all the way off and the button turned off for A/C It likely needs returned for an additional examination of the control module.

There is a compressor clutch relay in this vehicle. If the contacts are stuck, it'll power the comp clutch full time. Then the evap coil will freeze and the pressure switch will shut it off. Find the relay and yank it. If the comp shuts off, swap in another relay. If the problem still exists, you probably have a short to ground on the control coil side of the relay.

If the defroster is on the AC is usually on - this may be why you are hearing it run.

My 2001 Dakota AC works great when I turn it on, it has a new pump and had certified mechanics evacuate and recharge the system. But I can hear the pump cycle and engage even when I have the AC off, is this normal? I am concerned about fuel consumption and wear and tear on the pump.