Electronic – How to tell what type of capacitor to use with an electric motor, if any

I haven't even bothered watching after "only DC motors can be used as a generator".

As far as I am aware, a motor can be of the following families:

1. Permanent magnet DC brushed. DC back emf.
2. Coiled stator DC brushed (as a separate winding, or internally wound as series or parallel). DC back emf IF the stator is powered. Single phase universal motors are a subset of series connection types for which, regardless of the polarity of the voltage, torque is always generated (needs moveable brushes or a different wiring to change direction though).
3. Permanent magnet AC synchronous (three phases). Three phase AC back emf.
4. Coiled rotor AC synchronous. I think those generally are not brushed but rather rectify the current induced by the stator. If brushed, no back emf unless the rotor is powered.
5. DC brushless. This one is basically a permanent magnet AC synchronous with hall sensors built in, to be able to electronically switch the phases. The back emf is however square or trapezoidal to maximise flux linkage.
6. Stepper motor (2, 3, 5 phases). Close to the PM AC synchronous in its construction, except that the motor is made to maximise the number of stable equilibrium positions of the rotor (many alternating magnetic poles at the rotor or variable reluctance). Back emf depends on how it's driven.
7. AC asynchronous (3 phases). The rotor is a closed loop (a coil, or a squirrel cage made of bars) which creates its field from currents induced by the stator. Can only be used as a generator beyond the synchronous rpm (+voltage at stator). AC back emf (TBC).
8. AC asynchronous (single phase). The motor cannot be self-started unless an out-of-phase auxiliary supply is created via a reacting capacitor, and fed to windings 90° from the main windings. Can only be used as a generator beyond the synchronous rpm (+voltage at stator). AC back emf (TBC).

There are many more (e.g. hybrids), but I think those represent 95% of the production. I'm sure I've missed a few important ones, please feel free to comment and I'll update the list.

The biggest clue to the type of a motor is the number of wires, but as you can see this is not enough. Some motors cannot generate power without an excitation, some not at all, and even if they do, the back emf is funny sometimes (trapezoidal for example) depending on its construction.

You could plan to try the various types of supplies on the motor, ramping up the voltage, and see if it does anything, but what's your "OK that's not it, better cut the power before I smoke it" point? If you don't know what type of motor it is, I assume you don't know anything about it. Including the voltage and current ratings, Max rpm. You could get that from eyeballing it, but there is no guarantee then.

For your specific problem though, if you are certain your motor is a DC bruhless but you don't know if the inverter+control circuit are integrated, look at the number of wires. Generally the motor does not have a circuit built in, and an ESC must be connected to it. You will have to identify which wires are the hall sensors.

ESC might or might not be used for current generation, it depends on how they are made. I don't think there can be any harm in hooking up a resistive load compatible with its current range at the input and test it.

R in the R-C filter is the output impedance of the opamp driving your virtual ground, at the frequencies of interest - the noise frequencies.

At low frequencies it'll be very low (open-loop output Z / open-loop gain, assuming the opamp is used as a unity gain buffer) but it'll rise as gain reduces with frequency. Link to opamp's datasheet for more help.

Looking at your previous question, you may simply need to decouple the opamp input ... in which case R is simply (R1 and R2 in parallel), where R1 and R2 are the voltage divider on the opamp input.