The leaders of the Temple were the Kohanim (High Priests).
If you're referring to modern synagogues, the majority are run by an elected president with the spiritual leadership being the Rabbi and prayers being led by a Cantor.
The Kohanim (High Priests) ran the Temple. To clarify though, the Kohanim didn't lead the Jewish people.
The kohanim, who in turn were supervised by the Sanhedrin (court of Jewish Torah-sages).
The Kohenim (priests), however, they weren't leaders of the Jewish people, they were in charge of the Temple.
The Jewish temple in Jerusalem.
Jews worship at synagogues.
Jerusalem was the only place in the world that had a Jewish Temple at that time.
The Israelite/Hebrew/Jewish Temple in Jerusalem was destroyed by the Romans in 70 CE. It stood on the place called the Temple Mount / Har HaBayit, of which the Kotel ("Western Wall", "Wailing Wall") is the only surviving component. So there is presently nothing in the 'temple of Jerusalem', as there has been no temple of Jerusalem for over 1900 years.
Your question is ambigious. Jesus said our body is His temple.
The empire that took destroyed the First Jewish Temple in Jerusalem (and defeated the Southern Kingdom of Judah) was Babylonia in 586 BCE. The empire that destroyed the Second Jewish Temple in Jerusalem in 70 CE, after defeating the Jewish Zealots was the Roman Empire.
Chanukah (Hanukkah)
It is probably named after the residue that was taken from the Jewish Temple into the Valley of Hinnom, where it was burned.
The First Temple was located on the Temple Mount in Jerusalem, the same location as the Second Temple.
Judaism does not have a Holy Temple. It was destroyed in the year 70 by the Romans. In Jewish writings of today, it is simply referred to as the Temple in Jerusalem, the Holy Temple, or Beit Ha-Mikdash (Hebrew for house of holiness).
Nowhere that I know of.