In one Fatawa it said that the breath is blown into the fetus after 120 days. How do you calculate that? If somebody has an abortion before that specific time (120 days), is it lawful? When is abortion lawful, like having too much pain and some other health problems that are not vital but cause serious discomfort for the carrier?

Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

The life is infused into a fetus after having passed one hundred and twenty days from the beginning of the pregnancy.  Ibn Masoud narrated that the Prophet (Sallallahu Alaihi wa Sallam) said:  Allah's Apostle, the true and truly inspired said, "(as regards your creation), every one of you is collected in the womb of his mother for the first forty days, and then he becomes a clot for another forty days, and then a piece of flesh for another forty days. Then Allah sends an angel to breathe the soul into the body….

As for the abortion please read the Fatwa: 83891

Allah knows best.