Hello Brinda:
Your understanding of Nf governor function, as you explained it above, is based on two incorrect assumptions. The first error is your assumption that the Nf governor operates at a fixed Np value, and your second error is your statement that the Py valve on the Nf governor only has two positions, open or closed.
The Nf governor on a PT6A engine, such as the -67D that is the subject of the discussion here, operates at a variable Np. In forward thrust, that is typically 6% above the speed that the pilot has selected using the propeller lever. For example, if the pilot has selected 100% Np using the propeller lever, the Nf governor will begin to restrict fuel flow when Np reaches 106%. If the pilot has selected 80% Np, the Nf governor will begin to restrict fuel flow when the Np reaches 86%.
In forward thrust, the Nf governor serves two functions: First, it provides a fallback method of regulating Np in the event of a failure of both the primary and overspeed governor, and second, it provides a method of regulating propeller speed by reducing Wf in the event of, for example, a high-speed dive during which airflow over the propeller causes the propeller speed to grossly exceed that which the pilot has selected using the propeller lever.
In reverse thrust, the Nf governor serves an entirely different purpose - it prevents actual Np from ever reaching the speed selected by the pilot with the propeller lever. In reverse thrust operations, the propeller must always be kept in an underspeed condition relative to the primary governor. If Np was allowed to reach the speed set with the propeller lever during application of reverse thrust, the primary governor would begin to govern - by restricting oil supply to the propeller - and the reduction in oil supply to the propeller would drive it from a negative blade angle (reverse thrust) to a positive blade angle (forward thrust) - not at all what you would want when you have the power levers yanked back to row 3 in an attempt to stop on a short surface.
To accomplish this function during reverse thrust, the teleflex arm that leads from the cam cluster at the rear of the engine (near the FCU) to the outboard end of the reversing lever at the front of the engine has a small attachment on it called a 'Nf reset arm'. When the pilot pulls the power lever aft of the idle position, the Nf reset arm lowers the actuating mechanism for the Nf governor and recalibrates the cut-in speed of the Nf governor, from +6% of selected Np in forward flight to -5% of selected Np in reverse operations. These figures are typical, and may vary slightly depending on the airframe the engine is installed in, but the theory is true for all PT6A engine that are equipped with single action reversing propellers.
Because the Nf governor is reset to 5% less than the propeller speed selected by the pilot using the propeller lever in the flight compartment, the reversing propeller is prevented from ever reaching the speed at which the primary governor would kick in and attempt to regulate propeller speed. This ensures that the propeller is always kept in an underspeeding condition during reverse thrust operations, and thus control of propeller blade angle during reverse thrust application always remains with the beta reverse valve - and, by extension, with the pilot, whose hand on the power lever is ultimately connected directly to the beta reverse valve, via the power lever linkage, the cam cluster, the teleflex cable, and finally the reversing lever itself.
This also explains why the limitation given for Np during reverse in most AFM's is typically 5% less than normal maximum propeller speed. This limitation has nothing to do with any inability of the propeller to cope with a higher speed in reverse thrust, instead, it has everything to do with keeping the propeller in an underspeed condition, and keeping the primary governor 'out of the loop' during reverse thrust operation.
From this, I think you can see that we have established that the speed that the Nf governor kicks in at is not a fixed integer, it is a percentage value either higher (forward flight) or lower (reverse thrust) than what the pilot has selected using the propeller lever. Now you can also comprehend why the propeller lever must always be placed in the maximum RPM position before you can select reverse thrust...
Respecting your second point, about the Py bleed valve (that being the part of the Nf governor that controls Wf, or fuel flow) being either open or closed, that is incorrect. Py air pressure bleed invoked by the Nf governor will be proportionate to the requirement to reduce propeller speed. You can easily check this for yourself - go to your aircraft, look up the limitation for Np in reverse and take note of it, then slowly bring one engine all the way back into reverse until you notice that Np begins to govern at the published limit. It will be governing because the Nf governor is bleeding off Py pressure, thus reducing Wf as needed - no more and no less than is needed.
If you hog the power lever roughly back to the stop, you will certainly encounter momentary oscillations before Np (and, eventually, Ng as well) stabilize. But, if you slowly pull the power lever back until the reverse Np speed limit is reached, you will see that the Nf governor will smoothly and gradually begin to govern Wf and by extension Np, and both values will stabilize.
We see surging in Wf and engine temperatures when the Nf governor kicks in during heavy reversing after we land our aircraft because of external variables - changing airspeed, unstabilized propeller blade angles, and dynamic (torsional) movement of the propeller blades themselves. In addition to that, during normal landing operations, we rarely leave the power in the full reverse position long enough to allow Np, Wf and Ng to stabilize. But, if you conduct the above experiment slowly, you will see that the Nf governor has the capability to very smoothly modulate Wf to achieve the Np limit it has been set to maintain.
I hope this clarifies operation of this commonly misunderstood control mechanism.