What is the ruling on someone who entered the mosque while the Imam was in the second raka'ah, and feared missing the prayer time, so he bowed before the Imam to catch a raka'ah within the time, then completed his prayer?

This question is unclear, and if what we understood is correct, that you entered ihram (definition) individually and prayed the first rak'ah of Fajr, then joined the Imam in the second rak'ah, then your prayer is invalidated according to many scholars, while others deem it valid. The Shafi'is recommended that whoever begins an obligatory prayer individually and then wishes to join a congregation should complete their prayer as two rak'ahs with a taslim (definition), making it a supererogatory prayer, and then join the congregation. They disliked that one should continue their prayer with the intention of joining the congregation, and there are two opinions regarding its validity, the soundest of which is that it is valid. What you did is contrary to the Sunnah (definition), for the Sunnah is to join the Imam in whatever state he is found. And if you join the Imam and he then bows after sunrise, your prayer is not considered a make-up prayer (qada'a).