* As this is a day of mourning on the day of the birth of the same man, a sort of Tisha B'Av of the winter, as on Tisha B'Av, studying on this day is not allowed. Why, then, did the Hasidim refrain from fasting and wearing sackcloth and ashes? Maybe for the same reason that they kept Nittel a hidden secret: for fear of the anger of the Gentiles.
*Jesus, so it is told in the Sanhedrin tractate, was one of Rabbi Joshua ben Parchia's students. The prohibition of studying is also meant to prevent the remembering of his right to study on that day, and in order that his soul does not gain a speck of transcendence after death.
Herzl didn't say. Over the years, the Hasidim developed a genre of Nittel jokes. For example: A Hasidic rabbi was asked to eulogize Herzl. After thinking on it briefly, he came up with three virtues: Herzl never spoke when putting on tefillin, he never thought about religious law in dirty places, and he never dealt with the Torah on Nittel. Another joke tells the story of a man who was asked why he does not stop studying Torah on Nittel Nacht. "I keep the Nittel according to the Armenian Christmas," he answered.
Orthodox Nittel. The fact that the Greek-Orthodox and the Russian Church celebrate Christmas on January 6 causes some confusion among the Hasidim. It turns out that this doesn't entitle them to two Nittels, and the instruction is that everyone should mark Nittel on the same day that Christmas is celebrated in their country of origin. The Galicians do this on January 6. The Belz Hassidim do it on January 5, and it is not clear why. In the U.S., according to the decision by the Lubavitcher Rebbe, Nittel must be marked on the night between December 24 and 25.