It's faucet (which comes from middle Fr., whereas "tap" is English), Red. Fawcett is a surname.

A cock is, amongst other meanings, a device for regulating liquid flow. As a verb, most of its meanings revolve around "to turn, tip, or stick up". I reckon that's influenced or inspired the noun. In most/all cases a cock is a simple on/off, up/down device. So it's either turned up or turned down - 'cocked' one way or t'other.