Is there any symbolism or other explanation as to why there were specifically eight kings of Edom who ruled before any king in Yisrael did, like it says in Bereshis 36 (31-39)?
At the simplest level, it's saying that when Esav chose to sell off the birthright, he had the opportunity to fast-forward history -- boom! Monarchy, nationhood, no muss no fuss. Jacob, on the other hand, had to go on an arduous journey and process of becoming a nation. Thus, "Esav had eight kings while Israel still hadn't gotten to that stage." (If you want quick and easy, Judaism's not for you.)
But this doesn't say anything about the number 8 per se.
The first 8 kings proper of Israel and then Judah were:
- Yehoram (of Judah)
The 9th king was Ahazya, the one killed by Yehu. And after that was Queen Ataliah.
The first 8 kings were all righteous, and then Ahazya and Queen Ataliah were wicked.
Remember the prophecy / blessing from Yitzchak "when you feel aggrieved, you will break the yoke.." Once the kings of Judah turned wicked, Edom could break the yoke and rebel.
This is based on Rashi's intepretation.
There is an interesting commentary that Edom had 8 kings before Israel had even one because Jacob humbled himself before Esau at the time of their reunion and called him "my Lord" (adoni) 8 times.