To what extent is it correct to command the recitation of the Adhan in the right ear of a newborn and the Iqama in its left ear, and what are the benefits resulting from that, and is there a time limit that should not be exceeded for performing this act?

The Sunnah is to say the Adhan in the right ear of the newborn, as the Prophet, peace and blessings be upon him, said the Adhan in Al-Hasan's ear. As for the Iqamah in the left ear, no authentic hadith has been reported concerning it, though it was narrated from Umar ibn Abd al-Aziz. The benefit of this is to seek blessings and for the first thing the newborn hears to be the remembrance of Allah, so that his heart may be at peace. It is preferable for this to be done immediately after birth, or whenever it is convenient.