Starting; depending on aircraft. boost pumps maybe stby or always on. the boost pumps are to supply fuel to the high pressure fuel pump to avoid cavitation, and ensure good supply. If you have EDBP's you either select them off after start or it maybe done automatically.
On the PT6 the p2.5 ie: axial-centrifugal compressor has a bleed to avoid compressor stall, the centrifugal tends to move air at the square of the speed, so at low rpm not very efficient.
Takeoff; the PLA on the PT6 sets the N1 governor, adding fuel to bring up to scheduled engine speed. Normally on a PT6 N1 on takeoff will be between 92-95%. As you climb to Maintain the same torques N1 will increase to maintain the same gas motive flow
The Gas generator supplies motive flow to the power turbine. I like to say think of it like a torque converter on an automatic transmission, but sadly most people have torn apart a car's transmission to make the connection.
In Cruise, you normally set to a given temp as the engine will not normally be making 100% torque. Depending on the flat/derate power, most SOP's will have a max torque, N1 RPM, or Temperature.
WRT the MFP it runs whenever the engine is turning, and there is always some return bypass fuel flow, if the shaft shears on that, the engine stops......
"To maintain the correct air/fuel ratio for combustion, a lower fuel flow rate is therefore required." This isn't quite correct, the "mixture" doesn't count if you supply fuel it will stabilize at whatever speed the combined fuel/air mass flow will allow. IOW Self regulating.
This applies to the PT6, the Garrettt is a different Kettle of fish.
Basically in the Garrett, you have an underspeed governor, an overspeed governor, and a manual flow valve. The underspeed governor keeps the engine turning at 65%ish at idle on the ground, and 96-7% in the air. The condition levers set PCU RPM, and underspeed governor. as you apply power levers you are selecting FF, the Props will adjust to maintain set RPM 100 t/o, 97 in cruise. The overspeed governor kicks in if you exceed the upper limits.