This setup should not pose any problems so long as the servo tester, ESC, and BEC are made for the voltage you are giving it.
With most devices, you just have to worry about supplying the correct voltage because the device has an internal resistance which will cause it to draw a specified amount of current. This is okay because the manufacturers designed it that way.
A good example is with a lightbulb. If you directly connect 110V from an electrical outlet, it will only draw a very small amount of current. It has the capability of drawing thousands of amps, but it only draws couple hundred milliamps. This is because there is an internal resistance in the bulb, such as many other devices out there, that cause it to draw only so much.
Can I simulate this motor by using the equivalent circuit I draw above.
No. You need to use an equivalent circuit based on the Steinmetz equivalent circuit. For one phase of a 3-phase motor, the rotor and stator are each represented by an inductor and series resistor. The rotor and stator are connected by an ideal transformer. There is an inductor and resistor in parallel with the transformer primary winding to represent the magnetizing inductance of the stator. The load is represented by a variable resistance in series with the secondary circuit. The load resistance is inversely proportional to slip - infinite resistance at zero slip. The circuit is simplified by eliminating the transformer and adjusting the secondary component values accordingly.
For a single phase motor, the circuit is a bit more complicated.
Any suggestions on auxiliary winding, primary winding and capacitor values of a typical fan of this kind? I couldn't find any values of these given.
You are unlikely to find any useful data. You would need to preform dc resistance, no-load and locked rotor tests to get data to calculate the component values.
Is this a huge undertaking?
This is a substantial project. You can easily find on the internet step-by-step instructions for performing the tests and calculating the equivalent circuit values for three-phase motors. You can probably find the information for single-phase motors also, but it will be more difficult to find and likely more difficult to understand.
If you must have a good understanding of AC circuit analysis and electric machinery to do this.
Best Answer
It sounds like what you're wanting to do is to roll your own onboard ESC as part of your quadcopter design to save weight instead of using a prepackaged ESC on its own PCB. That's a perfectly reasonable thing to do, but there's a little more involved than just the microcontroller.
Most importantly, you need a power driver for each motor, either a dedicated BLDC driver IC or 6 discrete MOSFETs. If you use a microcontroller with enough PWM outputs, you won't need a separate BLDC controller IC in addition to the driver. Off the top of my head, there's STM32F0, PIC24, dsPIC, and LPC4350 chips that have dedicated precision motor control PWM timers.
I can't see a datasheet/wiring diagram for the motor you linked, but it has 3 wires (obnoxiously, all black) coming off of it, which means it's likely a standard 3-phase BLDC with no hall effect sensors built in.
The term you want to Google is "sensorless BLDC control", which will come up with a whole bunch of application note PDFs from various microcontroller vendors. This method of control requires use 3 channels (per motor) of your microcontroller's ADC to measure back EMF in the motor in order to tell when to energize the next phase of the motor.