Is تحية المسجد (two rak'ahs upon entering a mosque) obligatory after Fajr prayer at home? And what if there is not enough time to perform تحية المسجد before the Fajr prayer is established?

If you enter the mosque after performing the two rak'ahs of Fajr at home, then you should perform "Tahiyyat al-Masjid" (the prayer of greeting the mosque), due to the generality of the Prophet's ﷺ saying: "When any of you enters the mosque, he should not sit until he prays two rak'ahs before he sits." And if the prayer is established while you are praying Tahiyyat al-Masjid, then cut it short and join the Imam if you fear missing the first rak'ah; otherwise, complete it.