Is it essential with regard to the Sunnah prayers that are done before the fard prayer that the time for that fard prayer should have begun?

The time for a Sunnah
prayer that comes before a fard prayer does not begin until the time for
that prayer begins. 

Ibn Qudaamah said in al-Mughni (2/544) 

The time for every Sunnah prayer that comes before a fard
prayer lasts from when the time for that fard prayer begins until it is
done. The time for every Sunnah prayer that comes after a fard prayer lasts
from after the fard prayer is done until the time for that prayer ends. End
quote. 

Shaykh ‘Abd al-‘Azeez ibn Baaz (may Allaah have mercy on him)
was asked: 

I entered the mosque for Fajr prayer and prayed two rak’ahs,
and when I stood up for the second rak’ah, the muezzin began to give the
adhaan for prayer. I had intended that prayer to be the Sunnah of Fajr,
since when I left my house the adhaan was being given in some of the
mosques. When I had finished that prayer, I sat and read Qur’aan, and
someone next to me told me to get up and pray the Sunnah of Fajr. I told him
that I had already prayed, and he said, “That is not permissible, you have
to pray it again, because the muezzin gave the adhaan whilst you were
praying.” I hope that you can advise me about that. 

He replied: 

If the muezzin who gave the adhaan whilst you were praying
had delayed the adhaan and your prayer coincided with it after dawn had
broken, then you have performed the Sunnah and that is sufficient and there
is no need to repeat it. But if you are uncertain about that and you do not
know whether the muezzin who gave the adhaan whilst you were praying did so
after dawn had broken or when it was breaking, then it is more on the safe
side and better for you to repeat the two rak’ahs, so that you will have
done them after dawn for certain. End quote. 

Majmoo’ Fataawa al-Shaykh Ibn Baaz,
11/369, 370 

Shaykh Muhammad ibn Saalih al-‘Uthaymeen (may Allaah have
mercy on him) was asked: 

Is it essential with
regard to the regular Sunnah prayers that are offered before Zuhr and Fajr
that the time for the (fard) prayer should have begun? 

He replied: 

The regular sunnah prayers that are offered before a fard
prayer must be done after the time for the fard prayer begins. If we assume
that a person offers them before the time begins, thinking that the time has
begun, then he finds out that the time had not begun, then he must repeat
them, and the first prayer will be a general kind of naafil prayer, not a
regular Sunnah prayer. End quote. 

Liqaa’aat al-Baab il-Maftooh,
question no. 590 

And Allaah knows best.