Can a piezoelectric sensor detect constant water flow, or just the moment when the water is turned on and off?
Actually the answer is YES. Ultrasonic flow sensors which depend on piezoelectric transducers have been used for decades in precision flow measurement of gases and liquids. The principle usually requires a pair of transducers and electronics that that (1) excites a sound pulse across the fluid flow stream, and (2) receives the pulse. Doppler or time of flight principles are then used in the electronic circuits or software signal processing to determine the fluid velocity. And if you know the cross sectional area you can get the flow rate. You can read a more detailed account of ultrasonic flow sensors here.
EDIT: There is also another flow sensing technology that utilizes piezoelectric transducers - the vortex shedding flow sensor. For this technology a hydro(aero) dynamic interrupter is placed in the center of the flow stream, and is designed so that the flow streamlines separate at the back end of the interrupter - creating vortices. By placing a piezoelectric transducer at the back side of this interupter it can 'listen' to the vortex shedding. You can then count the number of vortices you hear and divide by the time interval to get a number that is proportional to the flow rate. You can learn more about vortex shedding physics here and the way Foxboro applies it to their product here.
This is a very vague questions, but the answer is most likely "No".
Piezo sensors produce a charge proportional to how much they are squished. That charge on a known capacitance results in a voltage, but at infinite impedance. Since infinite impedance is not practical in the real world, the steady state output of a piezo sensor will decay to 0 over time. Put another way, piezo sensors only show you the high pass filter of position, not position directly.
This means that piezo sensor are no good for measuring steady state. They work on changes. Therefore, a piezo sensor can't detect "constant" anything, including constant water flow, regardless of how that is connected.
If this "constant" water flow causes small vibrations due to turbulance or whatever, then you could possibly arrange a piezo sensor to detect it. However, note that this isn't really measuring "constant flow" as you asked about.