The pump only runs when you hit the "rest button" it uses the residual heat in the cooling system to provide heat with the engine off. In some cars it also circulated the coolant if the engine was turned off but very hot. I dont think yours was one of them
The default for the "mono" valve is open, that means it needs voltage to shut off, so coolant is flowing if you have a defective heater module that does not provide voltage for the closure. The logic would be, the faster you go the more air thru the system, the more heat. You could bypass the valve for a period of time