Assalaamu alaykum wa rahmatullaahi wa barakaatuhu. Do we reply to both azans of the Fajr (Azan of Tahajjud and azan of the Fajr)? Do we reply to both azans of the Friday prayer? Do we reply to the Iqaamah (second, small call to prayer)?

All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallahu ‘alayhi wa sallam, is His slave and Messenger. 

It is an act of the Sunnah to repeat the words of the athaan upon hearing it being pronounced. The Prophet, sallallahu ‘alayhi wa sallam, said, “Whenever you hear the mu'aththin, then say what he says.” [Muslim]

Hence, if you hear the two athaans of the Fajr or the two athaans of the Friday prayer, you should repeat what the mu'aththin is saying.

It is also an act of the Sunnah for the person who hears the Iqaamah to repeat it. The fatwa of the Permanent Committee for Ifta' reads:

“The Sunnah is for the one who is listening to the Iqaamah to say what the mu'aththin says because it is a second athaan. So you should respond as one responds to the athaan. When the mu'aththin says, 'Hayy ‘ala As-Salaah, hayya ‘ala Al-Falaah,' the listener should say, 'La hawla wa la quwwata illa Billah,' and when he says, 'Qad qaamat As-Salaah,' he should say the same, and he should not say, 'Aqaamaha Allahu wa Adaamaha (May Allah make it continually established)' because the hadeeth concerning saying this is weak, and it is narrated in an authentic hadeeth that the Messenger of Allah, sallallahu ‘alayhi wa sallam, said, ‘When you hear the mu'aththin, say what he says.’ This includes both the athaan and the Iqaamah, because both of them may be called athaan. Then he should send blessings upon the Prophet, sallallahu ‘alayhi wa sallam, after the mu'aththin says, 'La ilaaha illa Allah,' and he should say, 'Allahumma Rabba haathihi Ad-Da‘wati-Taammati was-Salaat Al-Qaa’imah, aati Muhammadan Al-Waseelata wal-Fadheelah, wab’athhu maqaaman mahmoodan allathi wa‘adtah.' (O Allah, the Lord of this perfect call and the prayer to be offered, grant Muhammad the privilege (of intercession) and also the eminence, and resurrect him to the praised position that You have promised), as he says after the athaan.”

Allah knows best.