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Yes, plugging a microphone into a headphone output could possibly damage the microphone.  Microphones are designed to work on tiny vibrations and tiny currents.  A dynamic microphone will work backwards as a speaker, but at very low power.

In any case, across all the various combinations of ouputs/inputs and plugging passive devices into them, it is quite unlikely that the electronics would be damaged.  The only case of damage I can think of is plugging in a delicate microphone into an output with significant power capability.

As for "headsets" versus "earphones", I don't know what the difference is supposed to be.  Supposedly both are passive devices intended to driven from audio outputs that are intended to source some power.