Hamlet's treatment of Ophelia, Claudius, and Gertrude

Hamlet

How do you react to Hamlet's treatment of Ophelia in Act 3, Scene 1? What evidence suggests that he knows he is being watched by Claudius and Polonius?

Hamlet shouts that there will be no more marriages, which is a direct threat to Claudius rather than to Ophelia. He also calls Ophelia false, and seems to believe that she has betrayed him, which only makes sense if he feels she is being used. While Hamlet's paranoia could be a product of his overwrought emotional state and hatred of Gertrude, the fact that he reacts so violently to her gentle rebuke suggests that he is angry at Ophelia for going along with what he sees as a trap, not just her romantic rejection.

What does Hamlet think of the Claudius's reaction to "The murder of Gonzago"? Why does Claudius not react to the dumb-show before the play-with-in-a-play?

Hamlet believes that the king's reaction is proof of his guilt. However, it could be that Claudius is more self-controlled, and does not react to the dumb-show because he does not wish to give himself away, and then he is simply is tired of seeing the Player King commit murder two times over, in Hamlet's production.

Why does Hamlet not kill Claudius when the king is at prayer?

Hamlet's states that he does not want to send Claudius to heaven. His father is condemned to purgatory, because Old Hamlet was not able to confess his sins, and Hamlet's father must walk the earth until he has done penance in the afterlife. Now Claudius is confessing and receiving absolution for his sins, Hamlet believes, so he should not send him "straight to heaven" while Claudius is at prayer.

Describe Hamlet's treatment of Gertrude during their confrontation in her private room? Is Hamlet justified in his treatment? Why does the Ghost appear here?