Identifying windings of a single phase induction motor

I believe you can use a standard wiring diagram from something like a Commander SK which anticipates what you want to do. I can't remember the circuit off the top of my head. I use them for three phase motors.

You can change the phase angle by changing the capacitor. I believe the torque will be very low.

I had to recently turn a single phase motor fan into a variable fan and we used a single phase chopper like is used to slow down a (series wound) drill. It worked fine and rattled a lot at low speed. That's all. The fan has a very low start up load and at low speed the magnetic field just slips a lot - no problem.

It works by "trying" to be synchronous but the voltage is not high enough so it slips and therefore slows down. You are far better off using the correct frequency so it tries to keep up, but can't because of low power, than to try to be synchronous and go at a low frequency/speed using a motor that is inherently unable to do that.

The only issue you will have with the former case is you may want to manually switch out the starter windings if it is running below the cut out speed. It might not be necessary. Just see if it starts to overheat.

Based on the video, I believe that you are correct. This is a permanent split capacitor motor. The capacitor is connected in series with the auxiliary winding to provide a phase shift to the current in that winding so that the motor has a defined direction of rotation and will start automatically. The best way to identify the auxiliary winding is to observe its series connection with the capacitor. It may also have slightly smaller wire. There are four main coils and for auxiliary coils in a 4-pole motor.