1) 13a & 14b - It's interesting that Yehoshua married Rachav Hazona (14b), and Calev married Basya bas Paroh (13a), even though these women did not have much Yichus! Any comment?
2) 13b - How could Esther immerse herself and then be with Mordechai,
without Achashveirosh finding out?
3) 13b - Why is Moshe Rabeinu's birthday more auspicious than his Yortzeit? We commemorate Yortzeits but not birthdays, I thought!
1) It is obvious that Rachav was a Tzadekes when she converted and married Yehoshua (and, as, the Gemara at the end of Ta'anis says, Yichus is not the only desirable attribute in a wife). Moreover, Rebbi Yerucham, in Sefer CHACHMAH U'MUSAR (1:7), explains that Rachav acted with tremendous Chesed towards the emissaries of the Jewish people, and for that she meritted to marry the leader of the Jewish people. In addition, since she acted with great Chesed on behalf of the entire Jewish people, she meritted that Nevi'im descended from her, who would serve the entire Jewish people.
He continues and explains that not only did she act with great Chesed, but she endangered her own life to do so, because she recognized the importance in saving the emissaries of Hash-m's holy nation. From here we learn the great merit that one has for working on behalf of the Jewish people, feeling pained when they are in trouble and feeling Simchah in their Simchah.
Also, the Sefarim ha'Kedoshim point out that great people of Yisrael come from forces which have their source in the Umos ha'Olam, in order to enable them to bring those nations to Avodas Hash-m (as we find that Shlomo ha'Melech married Bas Pharaoh, and Moshe Rabeinu married the daughter of the priest of Midyan).
2) Good point. It appears that the queen was not required to be in the king's palace at all times. The queen had her own palace. She was only expected to be with the king when she was called upon.
3) Chazal tell us that Hash-m fills the years of Tzadikim to the day; by being taken from this world on the same date on which they were born, it is a sign that a person's lifetime was complete and that he was a Tzadik, for he fulfilled his allotted lifespan. The day of his death, therefore, is the day that he arrives at a level of eternal completion, and thus there is no inauspiciousness associated with his day of death at all. The Gemara here means that Haman did not know that Moshe Rabeinu was born on the same date on which he died and that he had lived a complete life, a sign that his day of death was a good omen and not a bad one.
All the best,
Mordecai
As I read your answer regarding Moshe Rabbeinu, I recalled that the MAHARAL says something very similar, explaining that Moshe Rabeinu's life represented a "gilgul hachozer." (By the way, I believe the Chasam Sofer zt'l was also similarly bothered by the meaning of this gemorrah.)
Kol tuv.
That is correct -- the Maharal discusses this in OR CHADASH. He says that the fact that Moshe Rabeinu died on the date of his birth shows that he had attained Shleimus, as his beginning and end were bonded as one in time, like a sphere.
Mordecai