What is the ruling regarding a latecomer who performs the takbir while his Imam is rising from the first prostration, and when he descends to sit in a posture similar to the Imam's sitting, the Imam descends for prostration? Should he prostrate immediately, or should he sit like the Imam and then prostrate? And is his prayer valid if he sits and then prostrates?

It is recommended for a latecomer to join the Imam in whatever state he finds him, and to prostrate with him if he finds him prostrating, though this rak'ah is not counted. This is the practice among scholars. Your act of sitting before joining the Imam in prostration was an unintentional addition due to ignorance, and thus does not affect the validity of your prayer.