How long was the rivalry between burr and hamilton?
why does the author think burr wouldn't have been eager to kill?
how did hamilton's son phillip die?
why does the author think hamilton did not want to die in the duel?

I'm not sure about the first one but, 
Philip Hamilton was killed  in a duel by George I. Eacker.
Hamilton did not want to die because he and Burr were dueling in the same place Philip had died, and he did not want to leave behind the rest of his family, including his wife Eliza.
There's a lot of this info in the second act of the Hamilton Musical Soundtrack, if you need more help.

Aaron Burr and Alexander Hamilton had around 30 years of disagreements. Hamilton actually kept track of theses disagreements and made an itemized list which he later sent to Burr. Philip Hamilton died due to a gun shot wound from a duel with George Eacker. Eacker had insulted alexander in front of a crowd and Philip challenged him to a duel that took place across the Hudson river in new jersey. The bullet entered just above his hip and lodged in his right arm. As for the other questions, I am not sure.