Hamilton lyrical analysis question

spoken by eliza hamilton about alexander hamilton:
when he was ten, his father split, full of it, debt-ridden
two years later, see alex and his mother, bed-ridden
half-dead, sittin’ in their own sick
the scent thick
and alex got better but his mother went quick

eliza builds a few more character-building traits here. this is important because, again, the information was given by a third party member. what is the intended effect? (is it more or less believable?

what's the point of sharing multiple tragedies and hardships associated with hamilton's upbringing? consider how this contributes to the rhetorical power of the argument at hand, and - on a greater scale - for alexander hamilton?

why did eliza call hamilton alex?