If you rotate an electric motor by some external source, will it produce power?

Yes you are right. The basic principle of a generator is a conductor moving in a magnetic field. The voltage is proportional to the strength of the magnetic field, the speed the conductor cuts through this field, and the number of turns in the coil. The current generated in an electrical load connected to the generator causes a reaction through the magnetic field that requires more force to maintain the movement.

A motor is the reverse principle, where an electrical current causes a magnetic field around a conductor that reacts agains a second magnetic field, so the two can move and so move a mechanical load. Some sort of switching arrangement is required to make this a continuous action.

In principle all motors (including induction motors) can be used as generators, but the excitation can be tricky. These are generally called induction generators or asynchronous generators as opposed to the synchronous generator we usually call an alternator. Even a shaded pole induction motor like on small AC room fans generates a little AC when you spin it, presumable due to stray magnetic fields.

Yes a gear motor will, but keep in mind its made to do the opposite so it will not be as efficient as a generator, the same way a simple generator will spin when powered but not as efficiently as a motor would.

Yes

Very good question. Short answer is some do and some don't.