When is a rak'ah counted in congregational prayer, when is it obligatory to repeat it, and is the prayer of one who joins the Imam in the last prostration counted as congregational or individual?

If the latecomer catches the Imam bowing before the Imam rises, then he has caught the rak'ah. However, if he does not catch a complete rak'ah, such as catching the Imam in the final tashahhud, then he is considered to have caught the virtue of the congregation.