In order to reduce the length of Shakespeare's Hamlet, what scenes could be cut and what themes or issues might be affected in the play?

After having watched dozens of versions of Hamlet, I find it really interesting to answer this question.  There are a few scenes that are commonly cut and others that are greatly trimmed down in order to save time.  Most of the time, these scenes have to do with the minor characters (such as servants or friends) and do not have much to do with the main characters (Hamlet, Ophelia, the ghost, Polonius, Gertrude, and Claudius).  Sometimes these scenes have to do with the conquests and battles of Denmark.

Act 1, Scene 1 can be trimmed down.  There is a lot of conversation here between the characters on watch about the country's conquests.  The important part of this scene is the appearance of the ghost.

Act 2, Scene 1 can be trimmed down.  At the beginning of this scene, Polonius is asking Reynaldo to spy on Laertes in France.  Even though this event shows Polonius is fond of spying, it is not important to the main characters and could be cut.  The important part of this scene is Ophelia’s observation of Hamlet’s behavior.

Act 2, Scene 2 can be trimmed down with the news about the ailing King of Norway and other military decisions removed.  Everything else from this scene should remain.

Act 3, Scene 1 can be trimmed down.  The most important part of this scene is Hamlet’s interaction with Ophelia, so the length of conversation between the king and Rosencrantz and Guildenstern can be edited.

Act 4, Scene 2 could be completely removed in that it is about Hamlet simply speaking to Rosencrantz and Guildenstern about the disposal of Polonius’ body.  The scene does serve as a good example of Hamlet’s “antic disposition” because he obviously knows Rosencrantz and Guildenstern are spying here.

Act 4, Scene 3 can be trimmed down.  Again, the discussion of Polonius’ body is superfluous (with the exception of the worm comment made to Claudius).  What is important about this scene is that Hamlet is being sent to England to his death.

Act 4, Scene 4 could be completely removed as long as the director doesn’t mind removing the character of Fortinbras.  This is about the conquests of Denmark.  Even the comparison of Hamlet’s conquests and Denmark’s conquests are not necessarily important to the plot.  The only real importance is that Hamlet discusses his inaction.

Act 4, Scene 6 could be completely removed.  It is about Horatio and the sailors.  They do talk of Hamlet, but the scene becomes unimportant when Hamlet returns.

In regards to the second part of your question, the themes and issues affected here all have to do with the everyday doings of the Danish court and the country of Denmark.  Instead, most of the meat of this play  has to do with the doings of the main characters.  None of the main themes or issues would be affected.