Act 4 questions

Act IV, Scene 1
1. What is Claudius' main fear in the immediate aftermath of Polonius' death?
That he is next and going to die soon because of Hamlet.
Act IV, Scene 2
1. What does Hamlet refuse to tell Rosencrantz and Guildenstern?
Where Polonius' dead body is.
Act IV, Scene 3
1. What image does Hamlet use (ll. 19-29) to warn Claudius he's only king temporarily?
He says that killing a beggar is just like killing a king because they are both going to be dead and mean the same when there dead. 
2. Claudius ends the scene by writing a letter: to whom, and what order does it contain?
He tells the England King to kill Hamlet for him so no one will expect Claudius of planning it. 
Act IV, Scene 4
1. What's the value of the land Fortinbras' army is marching to capture in Poland (l. 20)? What will the invasion itself cost (l. 25)?
5 ducats, 20'000 ducats
2. Hamlet's soliloquy (ll. 32-66) is self-critical; summarize his main fault.
He is saying he is just a animal, because he just sleeps and eats. He states god gave him power to do something at least avenge his father. 
Act IV, Scene 5
1. Ophelia's songs during her first appearance in this scene deal with love, death and sex. Why? What do they tell us about her at the moment? What might they reveal about Her, Hamlet and Polonius?
She is stating that she misses him and loves him, but now she can control herself for the first time in her life because her father isn't there to tell her what to do, same thing with Hamlet.
2. Why is Laertes a danger to Claudius' throne (ll. 98-103)? (Actually two or three related reasons.)
Because he is young and wants revenge on his father's killer, second he will go at any lengths to achieve this goal, and the mob that was with him believes he should be king. 
3. What does Claudius offer as assurance that he had no part in Polonius' death (ll. 190-9)?
That if he is wrong Laertes can kill him and take the throne and all perks from that. 
Act IV, Scene 6
1. Horatio receives a letter from Hamlet explaining how he escaped from Rosencrantz and Guildenstern. How did he? 
He was attacked by pirates and they captured him, but now Hamlet owes a favor to the pirates.
Act IV, Scene 7
1. What reason does Claudius give Laertes for Hamlet's killing of Polonius (ll. 1-4)?
Because Claudius was friends with Polonius, Claudius says that Hamlet was trying to kill him not Polonius.  2. What are his two reasons for not charging Hamlet with murder (ll. 9-24)?
Because Claudius says the queen is Hamlet's mother, and he is married to Gertrude which is the queen. The public love Hamlet and would not want him to die or be in jail.   3. Claudius reveals that Laertes is famous for his skill with the rapier (a fencing weapon) and that Hamlet is envious of this fame.
Is this a question
4. How does Claudius plan to exploit this envy to give Laertes a chance for (publicly) guiltless revenge (ll. 126-38)?
He plans to what for Hamlet to come back, When he does people are going to place bets on Laertes to win a duel to the death. To avenge his father by killing Hamlet. 
5. How does Laertes refine the plan (ll. 138-147)?
He is going to poisin his blade so if he just scratches Hamlet he will die
6. What announcement does Gertrude make to end Act IV?
She makes the announcement that Ophelia had drown in a stream. (On purpose I think) She tells that she had flowers on her and she fell from a willow tree...weeping willow...