I know that Mi Yodea has already answered the question about the source of Maoz Tzur (Mordechai, unknown, 13th century, perhaps during Crusades?). Is it a universal custom or is it just an Ashkenazi custom to sing after candle-lighting?
It does not seem to be a universal custom, but nor is it limited today to Ashkenazim. R. Qafih's Teimani (Baladi) Siddur Siah Yerushalayim (2005) does not have it.
The Italian Siddur (Minhag Milano) published by the Del Mare family (1994) does not have it either.
Similarly, many Sephardi Siddurim do not include Maoz Tsur. These include Sukkat David published by Menahem Batsri (1986), Siddur Tsvi Yisrael printed with the approbation of R. Sh'lomo Amar, Siddur Tehillat Yitshak published by R. Eliyahu Tupik, and Siddur Yosef Hayyim (published by Siah Yerushalayim).[i]
However, it is found in some recent Sephardi Siddurim such as Avodat Hashem (1998) and Ohr Vaderech (1994).
[i] Additionally, R. Yitshak Yosef does not mention the recitation in his Yalkut Yosef, although does discuss recitation of hanerot halalu and mizmor shir hannukat habayit.