It's "their pursuit" and "their demise" (possessive)
"Americans, while occasionally willing to be serfs, have always been obstinate about being peasantry."
The quote (from Gatsby) illustrates the general desire of Americans to move up in class, just as Gatsby had the desire to move from a midwest farmboy to a Long Island socialite. Gatsby, because of his greed and lack of experience in the new society, ends up being alienated from the rich and blamed for their mistakes.
Honestly, not sure whether you've read it or not, but I would argue with your teacher that Hamlet's death was not a demise but a success because he accomplished his purpose in life. It could lead to some deep philosophical discussion.