Why did Hashem say "Ani Hashem"?
Rashi (in Va'eira): He taight Avraham that His Name was Hashem
See Rashi in Va'eira, Sh'mos, 6:9 DH 've'Gam Hakimosi'.
Why does the verse say "Hotzeisi'cha me'Ur Kasdim," which is a similar expression to "Hotzeisi'cha me'Eretz Mitzrayim" (Shemos 20:2)?
Chochmah u'Musar (Vol. 4, p.2): In order to be his E-lokim, He totally removed Avraham from Terach. So too for his descendants at the time of the Exodus.
"I am Hashem, Who brought you out of Ur Kasdim." Why does Hashem mention this here, just before the prophecy regarding the Egyptian exile?
Maharal #1 (Derech Chayim 5:4, p. 225): The beginning of the history of Am Yisrael would parallel that of Avraham, first of the Avos. Just as Avraham was thrown into the fiery furnace at Ur Kasdim, so were their sons thrown into the Nile in Mitzrayim. Just as Hashem miraculously saved Avraham in order to give him Eretz Yisrael, so too Am Yisrael.
Maharal #2 (Gevuros Hashem, beginning of Ch. 5): Both Avraham and Am Yisrael developed 1 in an environment that was the polar opposite of what they would represent in the world. Chazal say that Hashem regretted ever having created the wicked Kasdim (Sukah 52b), while Avraham justified the Creation of the entire world. Similarly, the Egyptians were profane and steeped in licentiousness, while the Jews were holy and chaste. 2
When opposites unite, they create wholeness. When the pure emerges from the impure, it teaches that they stem from the same Divine Source (Maharal, ibid. p. 29, 32).