What is the correct Sunnah when the child is born and how Azaan in a baby's ear should be performed?
You shall be answering this question for male.
It is a Sunnah to call the Athan in one of the earns of a new born,
رَأَيْتُ رَسُولَ اللَّهِ صلى الله عليه وسلم أَذَّنَ فِي أُذُنِ الْحَسَنِ بْنِ عَلِيٍّ - حِينَ وَلَدَتْهُ فَاطِمَةُ - بِالصَّلاَةِ
I saw the Messenger of Allah () uttering the call to prayer (Adhan) in the ear of al-Hasan ibn Ali when Fatimah gave birth to him.
As for the calling of the Athan in the right ear and calling the Iqamah in the left, it comes form a weak hadith, though scholars say that if you were to do such it is ok.
A newborn child should hear the Zikr of Allah upon their arrival into this world. After the baby is born the Adhan should be said in the right ear and the Takbeer (Iqamah) should be said in the left ear by anyone who is able to say the adhan of the Salah such as the father. This should be done as soon as possible.
Allah knows best.