The answer is in the lines immediately following: "There is a special providence in the fall of a sparrow. . . " What will augury - fortune-telling - tell Hamlet that will be of any use to him? This fatalistic attitude is foreshadowed in the gravediggers' gallows humor. Man is born to die; no one knows the hour of his death, but "the readiness is all."
Do you see how the passage shows a sense of resolution which is in contrast to his previous indecision? It is as if his adventure in England and his return home have changed him: from the melancholic to the frenetic (see his behavior towards Clausius after he kills Plounius) to calm determination. Only if Hamlet is prepared for his death by recognition of his nature and the nature of the universe can the play achieve tragic stature. If he dies unprepared, his death must be merely a pointless accident. That is not the stuff of tragedy.
Your other questions - Hamlet's characterization of Claudius, the Osric scene, and the Queen's speech, are separate matters.