Big variance. A flight instrument rotor spins at 10K to 15K RPM, according to Google. A spacecraft in low Earth orbit is at 1 rotation every 90 minutes. It doesn't matter.
If a toy gyro isn't rotating at high speed it falls over so it does matter.
This is not true. You can wiggle a toy gyro back and forth in your fingers,
Of course you can but that is just applying additional torques and muddying the waters of what is discussed here
The big difference between your last two lines is that one is 90 degrees and the other is 'about' 90 degrees - can you show me a gyro that doesn't precess at 90 degrees? That would be far more beneficial to your argument.
On a rotor, the mechanical input is the start of the process, the next stage is the aerodynamic forces that are a result of the mechanical change in pitch to the blades - these accelerate the blades, assisted by the mechanical input and governed by the laws of aerodynamics - the movement eventually cancels itself out when the lift produced is negated by the braking/damping effect of the air and the reducing mechanical input.