Is there any way to test my fuel pump to see if it is working? I have removed the tank from the vehicle to clean the fuel system and I what to check the pump to see if it works before I put it back in. The pickup screen is ripped so it definately needs replacing but I don't want to buy a new pump if I don't have to.

You can test if the pump is spinning if you connct it to a battery. However; just because the pump spins it doesn't mean it can produce adequate pressure & flow, also it is no good to power the pump in a dry condition.

Can I submerge the pump in water and just connect it to thebattery?

Thanks

Will the pump function in the car when the tank is empty, or is there something to prevent the pump from working on an empty tank?

It will work in with an empty tank, but only for a short whil, before getting damaged. Submerged fuel pumps use the fuel as a lubricant and a coolant.

What would happen to the fuel pump if I left the ignition in the on position with an empty tank, or kept cranking the engine on an empty tank?

Depends on the car. Some cars will shut down the pump within a few seconds if the engine fails to start. Other cars will allow the pump to run dry till it breaks.