Does "hearing" give off an electronic signal? I'm thinking in terms of the middle ear and the very intricate and tiny bones that process a "sound wave" into "what we hear." Is there an output to the middle ear ossicle chain bones such that a signal is being sent from the ears to outside of our body too? If so what is that frequency?
No there isn't any output for these bones other than mechanically pushing against the oval window.



In short:


Sound waves travel through the ear canal.
They hit the ear drum (Tympanic membrane).
The three bones (ossicles) in our middle ear are simply an impedance matching device (very much like gears on bicycles). They amplify the ear drum movement with the Stapes pushing against the oval window (part of the inner ear).
The mechanical movement of the stapes against the oval window causes liquid within the Cochlea to oscillate.
Within the Cochlea there are tiny hair cells that discharge (snap) in response to a specific frequency/amplitude (See this answer).
When discharge, a nerve pulse is sent to our brain.