Moses’ seat is in every synagog. It is today represented to us as the authority of the bishop, who has Moses seat, in a sense (the cathedra), the early bishop/pastors were like the presiders of the synagogs, as the faith grew in numbers the bishop/pastor would send elders to other locations and preside over them from a distance.
Jesus was a Pharisee. The other Pharisees regarded him as unorthodox (to the extereme)! But he was of the tradition that there is an afterlife, and he accepted the Prophetic books. The Pharisees had more influence in the Synagogs but did not dominate Judaism until after the destruction of the temple. The Council of Jamnia was a watershed event in Judaism, removing some books from the Old Testament (that seemed to support the teachings of Jesus) and basically establishing Pharisaic Judaism as a standard.
The Sadducees dominated the temple, they did not believe in an afterlife, this was a much older theological position in Judaism. The Sadducees (much like the Samaritans) were so conservative they were not accepting of the Prophets, and the books of the prophets. Their theological stance appears to have been a minority position, and they virtually dissapeared some time after the destruction of the temple.
The Sanhedin included Pharisees and Sadducees, they didn’t like each other but had to work together.