After watching a Munro video about it, I see your point. In the motor shown, the rotor gets its magnetic field simply by inducing a current and a field in it in reaction to the stator's field. There are no electromagnets in the rotor like I expected. In that case, I'm not sure either... I'd say more likely than not but it's complicated since the stator basically needs to induce a field and at the same time recover energy from the field that comes back from the rotor. I would further guess that the phase shift between the two components makes it possible to treat them separately.

Previous comment: Don't see why not - the "field" coils (the ones that replace the permanent magnets) need to be energized, which can initially come from the batteries if necessary.

Yes: IIRC some large generators work exactly like this, as the energized rotor gives a lot more flexibility in managing frequency and power output.