1. If you had a system were a water pump pushed water through a turbine, then the turbine ran 5-10 generators, how long could those generators power the pump before the system failed to work given that the water is refilled? And I'm talking on small scale like in your home or a science fair project, where you were using this to power something like a small lightbulb.

2.

3. How long depends on the size and nature of your system, but the simple answer is not very long.

Think of a dynamo that powers a light bulb on a bike wheel. Apply a spin to that wheel and probably the most you will get is 20 seconds of power.

4. What you propose would not work. The turbine requires energy to turn the generators, how much depends on the energy being drawn from the generators, how efficient all the components of the entire system are. There are always losses when energy is converted from one form to another, e.g. mechanical to electrical, along with friction losses. So the total output from all the generators is less than the energy required to operate the pump, and that is using the total output just for the pump.
"Perpetual motion" and "free energy" are not possible as they would violate the Laws of Thermodynamics.

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