Is it possible to make your own Du'a (I don't mean Dhikr) inside Fard Salah, for example: in Sajdah? Is it Sunnah or you need to do it after you finish making Salat?

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

The most correct opinion of scholars is that it is lawful to make Du'a in Salat asking Allah for the good in this life and the Hereafter if that does not contain any prohibition. This is the opinion of Shafi and Malik and it is backed by strong evidence. al-Nasa'i reported that the Prophet (Sallallahu Alaihi wa Sallam) said: "As for Sujud, (state of prostration) strive "during it" in making Du'a, it is indeed then that is more possible that your Du'a be answered" .
The Hanafis believe that the person should not ask for anything in his prayer that he could get from people, as such demands include the normal talk of people which is forbidden in Salat.
As for the Hanbalis they did not make Du'a lawful during prayer such as to get a beautiful wife, or a big house, or the like.
But the most correct opinion is what the Shafis and Maliktes believe.
Allah knows best.