Is my prayer valid if I pray the Sunnah of Fajr, then the Fard of Fajr without a Takbir, then the Sunnah of Fajr with a Takbir in succession? And is the Takbir obligatory before the Sunnah or Fard prayer? And when should the Fajr prayer be performed?

The correct method for performing the two rak'ahs of Fajr (Sunnah) is to perform them after ablution and ensuring that the time for the Fajr prayer has begun (the appearance of the true dawn). Then, two light rak'ahs are prayed, and they are valid with the takbirat al-ihram. As for the takbir mentioned by the questioner, if she means the takbirat al-ihram (Allahu Akbar), it is a pillar without which the prayer is not valid. If she means the iqamah, it is specific to the obligatory prayer. The time for the Fajr Sunnah begins with the appearance of the true dawn.