Why is it that basically every synagogue in the world has a menorah? One explanation is that on Chanukah there is a mitzvah to light candles in the synagogue as well as at home. But this explanation is problematic. Often the menorahs in synagogues have electric lights, or not the right number of lights. Thus they cannot be used for Chanukah.
A better answer may come from the parshiyot of the last few weeks. We read that there are exactly four items in the mishkan (and later the Temple): the ark containing tablets, the incense altar, the table with "showbread" on it, and the menorah. Three of these have close parallels in the synagogue. The ark, of course, is at the front of the synagogue, and like the Biblical ark it has holy texts in it and is separated from everything else by a curtain. There is no incense altar, but there is a square platform at the center of synagogue from which prayers are led and the Torah is read. And as previously mentioned, there is a menorah. I don't know why there is no "showbread" in the synagogue (understandable, since I don't know what the point of showbread in the Temple is either). But other than that, it seems there is a conscious effort to replicate the atmosphere of the Temple by putting all the "furniture" of the Temple into the synagogue. And for that reason synagogues tend to have otherwise functionless menorahs.
When I was about 8 years old it bothered me why, on Rosh Hashanah and Yom Kippur, the ark was opened for the most important parts of prayers. Was God located inside the ark? Were we praying to the Torah scrolls? I didn't receive a good answer, but as often happens with these things, I forgot my question and unconsciously accepted the ark-opening as "just the way things are". Until this last year, when my question suddenly came back to me for some reason. I asked a number of people and did not receive a satisfactory answer. But looking online, I found a source which gave me the first hints towards a solution.
It is indeed hard to justify opening the ark in a synagogue, if we regard it as simply a synagogue. But if the synagogue is a stand-in for the Temple, it makes much more sense. As my online source says, opening the ark is supposed to remind us of the Kohen Gadol's Yom Kippur entry to the innermost and holiest Temple chamber, where the ark was located. On the days on which we hope to approach closest to God, we reenact the historical occasion on which we came closest to God. Furthermore, the innermost chamber was not just a passive repository of holy documents. It was also the location from which God would "speak" and issue commands and prophecies (see Shemot 25:22). By exposing ourselves to the symbolic equivalent of the inner Temple chamber, we signal our receptiveness to these and other Divine messages.