When should the two rak'ahs of Fajr (Sunnah) be prayed if the worshiper enters the mosque and the congregational prayer has already begun: Is it after the congregation and before sunrise, or after sunrise?

If you catch the congregational prayer before performing the Ratibah (supererogatory) prayer of Fajr, then pray with them, and then pray the Sunnah (supererogatory) prayer of Fajr. There is no harm in that. Evidence for its permissibility is the Prophet's—peace and blessings be upon him—affirmation to Qais ibn Fahd—may Allah be pleased with him—when he saw him praying the two rak'ahs of Fajr after the Fajr prayer. It is preferable to pray them after sunrise, due to his—peace and blessings be upon him—saying: "Whoever has not prayed the two rak'ahs of Fajr, let him pray them after the sun rises."