1) He wonders whether he should make a stand (do you know a synonym?) against the mass of troubles and by doing so put an end to them. Hamlet is aware (realises??) that the end of the troubles can be brought about either by killing the king or by taking his own life.
How about -- He wonders whether he should bring an end to his mess of troubles. Hamlet knows that the end of these troubles can only come about by killing the king or taking his own life.
2) The idea of death as a sleep is a concept with which his audience was already familiar (can you say which...... with?)
In the Bible death is usually referred to as sleep (or as a sleep?=
Don't use "a" in front of "sleep" in these sentences. Also your phrasing with "with" inside the sentence is correct; to have "with" at the end of the sentence separates the preposition from its object.
3) He wonders whether there will be dreams after death. It is for this reason (is there a better way of expressing it) that men are so reluctant to take their own life and prefer to bear (would rather bear???) these earthly sorrows than fly to others they do not know of (is this correct?)
He wonders whether there will be dreams after death. This is why men are so reluctant to take their own lives and prefer to bear these earthly sorrows rather than to fly into the unknown. <~~??
4) The fear of something after death (can you say the doubt that there may be something after death), expressed by (or in??) the image of the undiscovered country from whose bourn (is boundary a synonym?) no traveller returns, puzzles him. It is their moral conscience, what (?) prevents men, and therefore Hamlet, from committing suicide (or prevents self-destruction ?)
Doubt that there is anything after death is expressed by the image of a place from which no one returns, and this puzzles him. Men's moral consciences usually prevent them from committing suicide. (Yes, that could also be ... from self-destruction. Your choice.)
5) It will take only a small knife to bring relief (is there a better way of expressing it??)
This sentence seems fine as it is.
6) He thinks of the insults which those who are good and quiet take (or the insults which are inflicted to those who ...) from those who are unworthy.
He thinks of the insults to which those who are good and quiet are subjected from those who are unworthy.
7) For my main course (or as my main course)I'd like (I'll have??)
8) Do you know a synonym for drop something on the floor (Ex Someone pushes you and let you drop something on the floor.
The word "drop" is fine. Here are some synonyms, but I don't see anything better: http://thesaurus.reference.com/browse/drop
9) Do you know a synonym for pour something over .... (Ex. He poured his milk shake over me
There are lots of good synonyms for "pour" -- http://thesaurus.reference.com/browse/pour -- "drench" seems really good, but so are some others.