Should I begin with the opening supplication (Du'a al-Istiftah) or directly recite Al-Fatihah if I enter the mosque while the Imam is already reciting?

If a worshipper enters while the Imam is reciting Al-Fatihah, he should say the takbirat al-ihram (opening takbir) and then remain silent until the Imam finishes, because he is commanded to listen attentively. This opinion is the soundest due to the strength of its supporting evidence in the saying of the Prophet, peace and blessings be upon him: "The Imam is appointed to be followed, so when he says takbir, say takbir, and when he recites, listen attentively." When the Imam finishes, the follower should recite the Du'a al-Istiftah (opening supplication), then seek refuge with Allah (say A'udhu billah), then recite Basmalah (Bismillahir Rahmanir Rahim) and Al-Fatihah. If he does not have enough time, he should skip the Istiftah and go straight to seeking refuge and reciting, due to the Prophet's saying, peace and blessings be upon him: "Do not do it except with the Mother of the Qur'an (Al-Fatihah), for there is no prayer for one who does not recite it," and his saying, peace and blessings be upon him: "There is no prayer for one who does not recite the Opening of the Book (Al-Fatihah)."

However, if the Imam is in a silent prayer, the worshipper should say the takbirat al-ihram, recite the Istiftah, seek refuge, and recite Al-Fatihah if he believes he can complete it before the Imam bows. Otherwise, he should immediately begin reciting Al-Fatihah.

If the Imam bows before the worshipper completes Al-Fatihah, the worshipper should bow with him, and the rest of Al-Fatihah is waived for him, due to the Prophet's saying, peace and blessings be upon him: "And when he bows, bow," and thus he catches the rak'ah.