Why does the Pasuk mention (a) Terach; and (b) Ur (lit. the furnace of) Kasdim, in connection with the death of Haran?
Rashi, Ramban and Targum Yonasan: Although the Pasuk might simply be informing us that Haran died in the lifetime of his father, 1 in a location called Ur Kasdim, 2 the Midrash explains that Haran was thrown into the furnace of Kasdim and killed, due to his father Terach, who had informed on his brother Avraham to Nimrod, because Avraham had smashed Terach's idols. 3
Kol Eliyahu: The Zohar says that Haran was the first (i.e. after the Flood) to die in his father's lifetime. Even though some of his ancestors died before their respective fathers, e.g. Arpachshad, Peleg and Nachor, verses prove that those ancestors were still alive when Haran died. Also refer to 11:11:151:2*.
"Ur Kasdim" may simply mean, "the valley of Kasdim" (as Rashi, citing Menachem ben Seruk translates it).
See Targum Yonasan for continuation of the story.
When the Torah writes that Haran died in Ur Kasdim, "the land of his birth," does it mean that Avraham was also born on Ur Kasdim?
Ramban: Not at all. 1 It was only Terach's youngest son Haran who was born in Ur Kasdim, after they had moved there. Avraham and Nachor were born in Charan.
In fact, that would be implausible, because (a) Ur Kasdim was situated in the territory of Cham (controlled by Nimrod), whereas Avraham was a descendent of Shem; (b) we see later (24:10) that Nachor, Terach's second son, who did not leave home together with Terach, Avraham and Sarah (in 11:31), lived in Charan, which is situated in Aram Naharayim (see Ramban, who elaborates in great detail).
Why does the Torah not explicitly mention Avraham's test at Ur Kasdim?
Ramban: The Torah would have had to write the claims of the idolaters, and it did not wish to.
Lev Eliyahu (Bereishis, p. 47): Avraham decided according to his intellect that it would be better to enter the furnace for HaSh-m's honor and leave this physical world, and ascend to bask under the Shechinah. There was no Satan (temptation) not to do so! 1 Even Nochrim must follow intellect, e.g. Derech Eretz. Torah requires overriding intellect at times, e.g. the Akeidah.
Perhaps this is why the Rambam does not list it among Avraham's 10 tests (PF). Rashi and R. Yonah (to Avos 5:3) do list it.
What is the symbolism of Avraham initially being persecuted by Nimrod, and later granted tranquility?
Maharal (Derech Chaim 5:4): Avraham endured suffering in his childhood, until he was designated by HaSh-m; this would symbolize the Jewish People, who suffered in Egypt until HaSh-m took them as a nation. 1
Also refer to 15:7:151.1:1.
