How to design an auto-adjusted equalizer?
To do this manually you need a pink-noise generator and spectrum analyser. Pink-noise is random noise with equal energy per octave - white noise has equal energy per decade. The microphone is positioned at the ideal listening position. The operator then adjusts a graphic equaliser to get the desired frequency response - usually flat.


  There will be two or more speakers which can also function as microphones.


This is unlikely to work. Speakers make poor microphones due to poor sensitivity and frequency response, speaker cable doesn't make good mic cable and the speakers aren't in the listeners' position.


  2- If it is valuable, what kind of components do I need?


A decent microphone positioned in the audience, a means of transmitting data back to the controller, software to interpret the data and send it to the equaliser device.


  3- Would it be reasonable to do equalizer-adjustment in real time?


Maybe, but you would need to compare the feedback with that transmitted and account for time-delay (due to slow speed of sound in the air), phase errors, etc.