1. How Was Yosef Allowed to Have a Haircut on Rosh Hashanah?
There are a number of different answers given:
i) The Moshav Zekanim answers:
The king commanded that Yosef needed to have a haircut, and it would have been pikuach nefesh [life threatening] not to listen. Therefore, Yosef was allowed to have a haircut, as pikuach nefesh overrides all prohibitions (besides for the three cardinal sins).
ii)
Yosef never cut his own hair, rather, he allowed a non-Jew to do it. And according to the Shach (Nekudas HaKesef, Yoreh Deah 198) in place of a mitzvah, amirah l’nochri [asking a non-Jew to carry out the forbidden activity] is allowed. Since Yosef was having a haircut for kavod hamalchus, he was allowed to ask a non-Jew to do it for him (Chasam Sofer, see Chasam Sofer in Shu”t Choshen Mishpot 190, where he says that kavod hamalchus is a mitzvah even for a non-Jewish king).
iii)
Presumably, Yosef fulfilled the halachah of having a haircut on erev Rosh Hashanah, therefore, on Rosh Hashanah when he was taken out of prison, he had nothing left to cut. He simply had another haircut due to kavod malchus, but since there was nothing to cut, there was no prohibition involved. (Shevet Yehudah)
iv)
The Ramban (Toldos) learns that the Avos only kept the Torah in Eretz Yisroel. Yosef was in Mitzrayim which is in Chutz La’aretz, therefore, he never kept the Torah there. We find that the Medrash on the pasuk, וטבוח טבח והכן, tells us that Yosef kept Shabbos, so we see that even in Mitzrayim he kept the Torah? However, the Ramban himself points this out and says that Shabbos was different to all other mitzvos. (Madanay Asher)
v)
Perhaps we can suggest, that Yosef was in a dark prison cell for many years and he had no idea what the date was, and whether it was Rosh Hashanah or not, therefore, he was unaware that there was a prohibition involved.
2. When Do We Find a Case That Because of the Obligation to Light Chanukah Candles, One Is Unable to Eat the Entire Night?
On motzei Shabbos Chanukah, if one doesn’t have enough money to buy both oil for Chanukah candles and wine for havdolah, then Chanukah candles come first (Orach Chaim 296:5). If one will have wine the next morning to be able to use for havdolah, then he must wait, and he shouldn’t eat before then (296:3). So, we have a case, that because of Chanukah candles, it is forbidden to eat the entire night (Tzavoh HaLevi).
3. Two Yidden Lit Menorah Without a Berachah, and They Both Realized Halfway Through Lighting That They Forgot, How Can It Be That One Can Still Make a Berachah and One Can’t?
The Mishnah Berurah (676:4) brings from R’ Akiva Eiger that if one started to light Chanukah candles without a berachah, and he remembered in the middle of lighting, one can still make a berachah on the remaining candles. If, however, the original candles went out, he can no longer make the berachah of “l’hadlik” (he can still recite “she’osah Nissim”). Therefore, the answer is: If two people forgot to make a berachah when lighting menorah, and they haven’t yet finished lighting, however, one person’s candles have gone out, and the other one’s hasn’t, the one who’s candles are still burning can make a berachah, however, his friend can’t. (R’ Zilberstein, Divrei Chemed)
4. How Can There Be a Case, That for the First Four Days of Chanukah, One Lights 1, 2, 3, and Then 4 Candles Adding One Each Night, and from the Fifth Day of Chanukah and On, He Lights 4,3,2, and Then 1, Going Down Each Night?
When Moshiach comes, speedily in our times, and we appoint a Sanhedrin, the halachah will be like Beis Shamai (see Siddur HaGra, on the berachah ‘Yotzer Ohr’). Therefore, if for example the Sanhedrin would be set up on the fourth day of Chanukah, from the fifth day and on the halachah will be like Beis Shamai, and we will light in accordance with his opinion, which is to go downwards every night of Chanukah (Mayim Chaim, Gilyon 515).
5. Why on Chanukah Do We Not Mention, “In the Days of Matisyahu ben Yochanan and Yehudis bas Yochanan”, Like We Mention, “In the Days of Mordechai and Esther” on Purim?
The Shevet Mussar answers: After Yehudis killed the king of Yovan, the miracles weren’t over and there were still many wars and kings to fight with. On Purim, however, through Esther, all the danger came to an end.
6. Why Did Yosef Emphasize Gratitude Over Theft When Accusing His Brothers?
Citing the Altar of Slabodka, Rabbi Chaim Zvi Senter explains that knowing his brothers and the upbringing they had received, Yosef understood that the worst offense with which he could charge them was a lack of hakoras hatov [gratitude]. Although stealing is certainly a crime, it is a sin that a person may struggle to overcome in a moment of temptation. In contrast, being ungrateful reveals that someone’s entire moral foundation is defective. We are called Yehudim because the name Yehuda connotes feelings of gratitude (Bereishis 29:35) that are part of our very essence. Particularly during Chanukah, which was established as a time to give thanks (Shabbos 21b), we should strive to reflect upon and express our appreciation to our parents, spouses, and most importantly, Hashem, for all they have done for us. (R’ Ozer Alport)