Should the pump run if the ignition is in the on position with the engine off?

Yes, the fuel pump should run when the ignition is in the "on" position with the engine off, but only for a few seconds. This allows the system to build fuel pressure before the engine starts. However, if the ignition remains on without the engine running for an extended period, the pump may shut off to prevent overheating and potential damage. Always consult your vehicle's manual for specific behavior related to your model.