Strictly speaking, a "synagogue" is a house of study. (Studying Torah is considered a form of worship.) We "belong" to a synagogue in order to support it, primarily financially. (Unlike Christians, we cannot take monetary offerings during the worship service, so there is no "collection.")
Certain required religious rituals and prayers can only be performed in the presence of a minyan -- a group of at least 10 Jews. And a synagogue is the most likely place to find a minyan.
as Synagogue is to Jew
Maybe, if it's a non-denominational synagogue.
No one works as a priest in a synagogue. Although Jewish men know if they belong to the priestly tribe today, there is no active priesthood in Judaism.
Yes.
Synagogue IS important to Jews.
Prayer, Torah study, discussing communal affairs, and weddings.
A synagogue is the place where a Jew preys to God. A synagogue is a temple, just like a mosque or a church. A synagogue is led by a rabbi (Jewish priest) and the service often involves preyers in Hebrew.
Sacred building That belong to a certain religion
No. Jewish custom is never to enter the house of worship for another religion.
Synagogues are places where Jews meet for worship and study.
How much a Jew worships in a synagogue varies greatly, and is not dictated by religion. Traditional Jews pray 3 times a day, but not necessarily in a synagogue. Jews go to a synagogue to assemble with other Jews and to preserve the Jewish community, in addition to prayer and study.
I'll assume you're referring to Hanukah. the menorah (candelabrum) is lit by each Jew in their home; but Hanukah is marked in the synagogue as well, with Torah-reading each day for the eight days, and a couple of additions to the daily prayers. It is customary to light a menorah in the synagogue too.