Is it necessary to interrupt the two rak'ahs of "Tahiyyat al-Masjid" (Greeting of the Mosque) or to skip the last rak'ah when the congregational prayer begins?

If you can complete Tahiyat al-Masjid (the prayer for greeting the mosque) before the first rak'ah of the obligatory prayer passes, then complete it. Otherwise, interrupt it and join the Imam.