I am in no way, shape, or form guaranteeing that any of these answers are correct. Through my own research, I compiled a list of what I feel the best, most correct answers are. Feel free to pick and choose, alter, or ignore any of the following :) I hope it helps!
Scene One
1. He doesn't necessarily admit to his crime, but to his guilt. He admits that he feels awful about what he's done, and hates having to hide his crime from the world.
2. Abuse from superiors (quite possibly resulting from Claudius' treatment of him), insults from arrogant men (again, probably referring to Claudius), Unrequited love (possibly referring to his mother's betrayal of his father by marrying so shortly after his death), and mistreatment of good people (referring to himself, his father, etc).
3. He portrays death as nothing more than a "big sleep." He develops this idea more by later going on to say how the only reason people fear death is because they're afraid of what they might dream about.
4. She reveals how much his character has changed-- he used to be very kind and loving, and very affectionate. This provides a strong contrast to his now bitter and reserved character.
5. Hamlet is blaming all women for causing his problems. He forgets that it "takes two to tango" so to speak, in that men are also responsible for his creation. There is some suggestion that, at the time, the word nunnery referred to a convent or a brothel, which may have contributed to the scenes ambiguity.
6. That women (such as herself, and more importantly, his mother) are too focused on ensuring their own sexual well being than they are on having a happy and healthy relationship. Basically, women are bad.
Scene Two
1. Hamlet wants to make sure they act the show out perfectly so that it will add to his Uncle's guilt a great deal more; it progresses the story through the eventual revelation of his uncle's guilt.
2. Horatio is, in essence, just a really good guy. Hamlet tells him that he's basically the best man he's ever met. He's not a slave to his passion (there's another sting at women, who he claims are).
3. He tells him everything that he learned from the ghost, and he wants his help to watch Claudius' reaction to ensure that his uncle really is guilty of what the ghost said he was.
4. He switches from iambic pentameter (representative of sanity and knowledge of social class) to prose (representative of madness and shirking standards).
5. He's mad at her for choosing sides against him, and wants to contribute to his portrayal of madness and vulgarity.
6. She is entirely unaware of Claudius' guilt, but at the same time, feels guilt from her own actions (ie marrying so quickly after her husband's death).
7. Hamlet compares himself to a recorder-- he gets sort of offended that many think they can play him that easily.
8. Nero murdered his own mother-- Hamlet promises to himself that he's not going to do that.
Scene Three:
1. It makes the king realize his own guilt, it confirms Hamlet and Horatio's suspicion as to his guilt, it affirms the reality of the ghost, it makes Gertrude realize her guilt at the situation, and it makes many people think that the king feels guilty.
2. Gertrude is no longer the queen, and is instead a mother and a woman. They view her as a such, and therefore, she becomes someone who can't be trusted because she might lie to protect her son.
3. He's trapped between his guilt over killing Hamlet sr. and his pride (he doesn't want to give up his title as king and his ability to rule for the sake of repentance-- he thinks that if he does repent, he's going to have to do so).
4. Because he just repented, and therefore he might not go to hell. Hamlet wants to make sure that when he kills Claudius, he'll go straight to hell for what he did.
Scene Four
1. Hamlet (knowingly or not) uses different interpretations or views on many words his mother uses and throws them back at her, further adding to her view of his as "mad."
2. He thought that he was killing Claudius, not Polonius. He's almost surprised, and a bit sad that it wasn't the king.
3. She feels guilty about marrying Claudius so soon after Hamlet's fathers death-- she feels as though it's betraying him.
4. The intimacy (aka sex). It's incest, and Hamlet struggles with the idea and thought of his Uncle putting his "filthy" hands on his mother.
5. It confirms to his mother how seemingly mad Hamlet is, because she can't see the ghost, so it appears as though he's speaking to thin air.