Green Eggs and Hamlet - Generative Questions

For 5.1 (four questions):

1. Why does the gravedigger sing (5.1.64-69, 5.1.72-77)? Is his reason the similar to or different from the reason Ophelia sings in Act 4?
2. What is Shakespeare's reason for putting the play on the word "lie" between Hamlet and the gravedigger 5.1.123-132)? Does it reveal anything to the reader or just confuse it further?
   
3. When Hamlet admits to loving Ophelia (5.1.285) Claudius calls Hamlet "mad", Why?
4. When Hamlet talks about how he loved Ophelia so much he would jump in the grave with her Gertrude says "this is mere madness" (5.5.302). Why is Gertrue so quick to deem Hamlet's behavior "mere madness"? Is she listening to Hamlet's previous advice to her, or is she just agreeing with Claudus?