Is a rak'ah counted for the worshiper who joined the imam during rukoo' (bowing), and the imam did not become aware of him until the second prostration, knowing that the imam did not intend to lead the prayer until the second prostration?

The prayer of someone who joins you is valid, and according to the most authoritative opinion among us, the intention of leading the prayer is not a condition for the validity of the emulation. This is the view of the Malikis, Shafi'is, and Hanafis, and one of two narrations from Ahmad, contrary to the Hanbalis. The view that the Imam's intention of leading is not a condition is the choice of Sheikh Ibn Uthaymin.