The Iqama or Iqamah (Arabic: إِقَامَة, ʾIqāmah) is the second call to Islamic Prayer, given immediately before the prayer begins. Generally, the iqama is given more quickly and in a more monotonous fashion, compared to the adhan, as it is addressed to those already in the mosque rather than a reminder for those outside it to come to the mosque. Aside from a difference in the number of repetitions of each formula.
It has started when the Prophet Muhammad (SAWA) declared Ali as his successor on his return from his last Hajj on the famous day of Ghadeer in front of the largest gathering in the history of Islam in his time. The companions who witnessed the declaration of Ghadeer were over 120 thousands up to 125 thousands according to historic sources. The Prophet asked all the companions who were present: Am I not more entitled to be obeyed by the believers, than themselves on themselves? They replied him: Yes, O Messenger of Allah. He said immediately raising the hand of Ali in front of all of them : So, for whom I am the Master, Ali is his Master من كنت مولاه فعلي مولاه. The Prophet then asked all the companions to give allegiance to Ali as Mawla (Master).
First Athaan (Azaan) after Ghdeer deceleration had in it ASHHADU ANNA ALIYYAN AMEERUL MO'MINEEN.أشهد أن عليا أمير المؤمنين
Some persons who were not happy with Ali did not like that and complained to the Prophet. He was angry with them and asked them: What did we say in Ghadeer?
This is narrated even in Sunni books like Al-Salafah Fi Akhbaar al-Khilafah by al-Maraaghi, a Well known Sunni 'Aalim from Egypt. Page 34. This book is in Al-Dhaahiriyyah Library in Damascus.
The Prophet Muhammad (SAWA) ordered the Muslims to always mention Ali as Ameerul Mo'mineen whenever they mention his name as the Messenger of Allah . (Al-Ehtijaaj by Al-Tabrasi). You can see in Musnad Ahmad Ibn Hanbal the Hadeeth from the Prophet Muhammad (SAWA) : When I was shown the gates of the Paradise, I saw that it is written on it : La Ilaaha Illallah, Muhammad RASOOLULLAH, Ali is the brother of RASOOLULLAH.
Yes she can. There is no condition in Adhan and Iqaamah to be given by the father only or any man. Women including the mother can do it as well.
'Giving the name to the child should not ignore the father's wish.