Why does the Torah find it necessary to tell us about the eight kings of Edom?
Rashi: In keeping with the prophecy (Bereishis 25:23) that 'One nation will be stronger than the other' (i.e. but never equal), the Torah records how the first to gain power was Edom. 1 Correspondingly, eight kings later ruled over Yisrael between the time of Shaul and Yehoshafat, to whom Edom was subservient, and it was only during the reign of the latter's son, Yoram, that Edom broke away from Yisrael and placed a king on the throne. 2
Ramban: To teach us the fulfillment of Yitzchak's Berachah, "And you will live by your sword," 3 when they overpowered Se'ir ha'Chori and took over his land. All the towns that are mentioned in the subsequent Pesukim were in Edom. 4
Moshav Zekenim: This shows like Chazal expounded, "Bazuy Atah Me'od" 5 (Ovadyah 1:2) - Edom does not establish a king, the son of king. Every king listed was from a different city than the one before.
Ohr ha'Chayim #1: The Torah teaches who [married into or] established a king of Edom, for all of them will be destroyed with Edom at the end of days.
Ohr ha'Chayim #2: Just like Edom's kingdom extended a long time, the kingdom of Yisrael will extend the same time. Even though the world is only 6000 years, Hashem will extend the days of Mashi'ach past this, like it says in Sanhedrin (99a).
Malbim (to Divrei ha'Yamim I 1:43): Edom was not able to establish a king from Edom itself, and not a king the son of a king. Every king listed was from a different land than the one before. Ha'amek Davar - Amalek had their own kings, but they are not the primary country of Edom called by Esav's name. Edom itself did not establish kings from Edom until after there were kings of Yisrael.
See Ba'al ha'Turim.
As the Pesukim testify in Melachim I 22:48, II 8:20.
So Rashi expounded on that verse. Avodah Zarah 10a expounds this from the beginning of that verse - "Katon Nesaticha ba'Goyim."
What is the significance of the statement, "before a king ruled over Yisrael?" To which era does this apply?
Ramban #2, Rashbam, Hadar Zekenim and Seforno: The king of Yisrael referred to here is Moshe Rabeinu, 2 and the two sets were indeed consecutive. 3
Malbim (to Divrei ha'Yamim I 1:43): It was until Yehoshua conquered Eretz Yisrael and was a king. From then, Edom did not have a king until the reign of Yoram. The 11 chiefs ran the country alone.
Ramban: According to this however, the two sets of kings were not consecutive, as there was a huge gap between the kings of Edom and those of Yisrael, and the Pasuk is only coming to tell us that, when Yisrael would have kings, Edom will not reign, in keeping with Yitzchak's prophecy, "and you will serve your brother" (Bereishis 27:40).
Hadar Zekenim, Rashbam: You cannot say that it refers to before Sha'ul, for Yosipon counts almost 40 kings of Edom between these and David (who was only a few years after Sha'ul). See also Targum Yonasan to Devarim 33:4.
QUESTIONS ON RASHI
Rashi writes: "And these are the kings [who ruled in the land of Edom] - There were eight; and Yaakov produced a parallel [number]." Why did Yaakov and Esav produce the same number of kings?
Gur Aryeh: Rivkah was told, "Two nations are in your womb" (25:23); i.e. Yaakov and Esav would parallel each other as rivals. (Externally, Yaakov and Esav were similar; refer to 25:23:151.1:1 and the note there.)
Rashi writes: "And these are the kings - ... Yaakov produced the same; and the kingdom of Esav was dormant in their times." Why couldn't kings of Yisrael and Esav exist simultaneously?
Gur Aryeh: Rivkah was told, "One nation will overpower the other" (25:23); and as Rashi explained, "When one rises, the other falls, as prophesied by Yechezkel (26:2)" (Rashi to 25:23).
