I recommend Fleming’s The Duel on these questions.
In general, however, Burr had no role in the Jefferson Administration because he omitted to make a public statement as the House considered the resolution of the presidential election of 1800 that Jefferson should be preferred over him. Although the two of them knew each other from their days as attorneys, and in fact had argued cases on the same side–and, if memory serves, even shared hotel rooms–Hamilton bad-mouthed Burr from early on in their relationship. The first time this led Burr to demand satisfaction (clarification or a retraction), Hamilton complied. When he didn’t the second time, Burr demanded a duel. I have Facebook friends in two different time zones who have told me, unbidden and without knowing each other, that they are Burr relatives and that their family tradition is that VP Burr should have shot Hamilton sooner.
He ran for NY governor because he was kicked off the Republican ticket in 1804.
Burr was charged with murder because dueling was murder under NJ law in 1804, as I recall. It seems that Hamilton did miss on purpose.
I think it was a sad, sorry, sordid affair. Under the code duello, Burr didn’t have a choice but to challenge Hamilton. Actually hitting him, however, was entirely discretionary.
Fleming’s really is a good book. The code duello is described in Joanne Freeman’s Affairs of Honor.