Is Fajr prayer the Ragheebah prayer, or does it have another name, and how is it prayed, in the mosque or at home?

First: What is meant by Fajr prayer or Subh prayer are the two obligatory rak'ahs. As for the two rak'ahs prayed before them, they are the Sunnah of Fajr or Subh, and are called the two rak'ahs of Fajr or "Salat al-Raghibah" due to their merit.

Second: It is preferable to perform the Sunnah of Fajr at home before leaving for the prayer, as authentic hadiths indicate that the Prophet (peace be upon him) used to pray them in his home, which is consistent with his saying (peace be upon him): "The best prayer is a person's prayer in his home, except for the obligatory prayers."