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Yes, ALL Orthodox synagogues do, but in the other synagogues, some non-Orthodox Jews will wear their own.
In my experience (in Orthodox synagogues) it is; but other kinds of synagogues might do things differently. In Ashkenazi Orthodox synagogues, the chazzan stands at the bimah only during the Torah-reading. Otherwise, he has a lectern which is at the front of the synagogue.
Because they have different beliefs.
It's mostly the Orthodox Jews that do that.
They're only separated in Orthodox synagogues, which make up a minority of synagogues in North America. Orthodox synagogues have separate seating for males and females in order to focus on the prayers instead of on each other.
Orthodox Jewish men go only to Orthodox synagogues. Non-Orthodox Jewish men would be likely to go to non-Orthodox places of worship. In Israel, Canada, the United Kingdom, and the United States, there are prevalent numbers of non-Orthodox or Liberal synagogues. As a result, people in those places who are themselves Liberal can often find a Liberal synagogue in which to worship. Outside of those four countries, it is very rare to find a Liberal synagogue, so Jews of all religiousities (both Liberal and Orthodox) go to Orthodox synagogues for prayer, but will not perform all of the Orthodox Mitzvot when not in synagogue.
Yes. But in Orthodox synagogues, only the men can dance.
In order to concentrate on the prayers, not on the opposite sex.See also:More about Jewish prayer and synagogues
Try the Orthodox synagogues.
In Orthodox synagogues, men and women are seated separately.
In orthodox synagogues, it's in the center of the room.
In Orthodox synagogues, most of the service is in Hebrew, except the sermon or teaching. In Reform synagogues, there's some Hebrew, but most of the service is conducted in the language of the land.