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I am not sure what you mean by "each side of the tap" but I will take position 1 to be at the 12 m above the tap and position 2 to be at...

In the first link there are several formulas for the pressure but I don't think using any of those formulas can result in P to be zero...

I would say it is possible. So if this situation happens, then the water won't flow even though the tap is open? Thanks

Approximately correct because the absolute pressure will be zero only at 0 K? Thanks

Assuming ideal gas and constant temperature, then as the water level drops, the volume increases so the pressure decreases. Not sure...