There is so much to see and say here, and no time to write it all down in a clear form. Here is some of it, anyway:
On the line, “Better to throw one’s self into an oven than to embarrass another person publicly,” Rashi takes our gemara simply: We see that one should let himself be put in an oven rather than embarrass someone else, since Tamar was willing to suffer this fate. Tosafos in Bava Metzia 59a points out that this is seen in the Torah’s pasuk itself, in which the word מוצאת does not have any vowels on the א, so that the word is actually מוצת, to be kindled (a la ויצת אש בציון – Eichah 4).
The gemara notes the recurrence of הכר נא in the book of Bereishit linking the sale of Yosef with Yehudah/Tamar, but there are many more borrowed phrases and images unifying Bereishit (as well as the Torah itself), and giving the lie to the Documentary Hypothesis. A few quick examples: The young goats (Yaakov’s meal for Yitzchak, the goat’s blood for Yosef’s tunic, Tamar’s fee), the link of עלי קללתך and עלי היו כלנה, the numerous references to walking in the דרך ה', and the cross-biblical theme of צדקה ומשפט which carries us from Bereishit 18:19 all the way through Devarim 33:21 and then into the neviim.
The gemara talks about Yehudah being named for his future admission/הודאה, but the Torah gives a reason for his name – it’s הפעם אודה את ה', Leah’s thanks to Gd! Maharsha says she would have named him אודה under her original reason; HaShem inspired her to call him יהודה.
I am puzzled by the gemara’s declaration that Yehudah sanctified the Name of HaShem by publicly admitting his wrong; we are taught אשרי נשוי פשע כסוי חטאה (and see the Rambam in the beginning of Hilchot Teshuvah on this point) that one should not divulge private sin to the masses, lest it actually cause a desecration and desensitization!
See the Maharsha on the gemara's analysis of the pasuk describing Miriam's wait for Moshe.
Why do we bring a pasuk from Yeshayah מי נח זאת לי as proof that won’t destroy the world with water, instead of bringing the post-Flood biblical pledge, לא אוסיף להכות את כל חי?! Perhaps because that earlier pasuk isn't water-specific?
The gemara here regarding Yisro's reward for refusing to participate in Paroh’s persecution indicates that Yisro's descendants, or at least some of them, did become Jewish. (See our discussion on Yael.) But then why did they live next to Amalek, per Shemuel I on Shaul’s war with Amalek?
Some have the minhag of blowing a Teruah Gedolah at the end of Yom Kippur. This mirrors Rashi here, that the Jews heard a Teruah at Har Sinai, since the shofar blast at the end of the Revelation at Har Sinai (במשוך היובל המה יעלו בהר) is one of the sources for the shofar blast at the end of Yom Kippur.
Rashi's two approaches to translating the gemara on ערי מסכנות take the Gemara’s line in opposite directions – one is that it's about the building in Egypt, the other is that it's about construction in general.
On “resembling thorns in their eyes,” the Maharsha's approach (the Egyptians resembled thorns in their own eyes) seems to fit the wording more accurately than Rashi (the Egyptians felt punctured by thorns).
The pasuk brought here, תחת התפוח עוררתיך שמה חבלתך אמך שמה חבלה יולדתך, is one source for the myth that the fruit in the garden was the apple, from a mis-reading of the Hebrew root ח-ב-ל as corruption – “Under the apple tree, your mother corrupted you.” It should be read like חבלי לידה, “your mother birthed you.” (There is also a second source, the Latin “mal” which is associated with the apple/malus.)
The Maharsha explains why I might be more or less likely to identify Miriam or Elisheva as the second of the meyaldot.
The Gemara’s derash readings of חיות depend on reading it with a patach under the ח, instead of the actual kamatz.
The gemara that says Miriam gave birth to Chur after her illness must not be referring to her illness with tzaraat, but rather to a childhood illness, for Chur was dead (per midrash) by the time she experienced her tzaraat. This also fits 12a; see Rashi on 12a עזובה.