In Act 2 scene 2 of Hamlet, what thematic significance do the players have to the events in Denmark?

Shakespeare loved to use the theatre and theatrical
conventions to create a mirror to real life.  Of course, to a large degree, that is the
point of the plays themselves -- to cause the audience to witness the events that take
place on stage and then turn their gaze inward and see the same dilemmas confronted by
the characters on stage mirrored in their own hearts.  In Hamlet,
this theatrical device is made literal when a group of traveling actors actually appear
as characters in the play.  They relate to a very common theme in Shakespeare's works --
the theme of appearance (or pretense/illusion) versus
reality.

The most obvious way that this theme reflects the
events of Denmark is that Hamlet creates a play for the actors to perform (illusion)
that closely mirrors the events described to him by the ghost of his father (reality). 
He plans to have the players perform this "Mousetrap" in order to see if Claudius reacts
in a way that confirms his guilt.  In this way, Hamlet will use "appearance or illusion"
to confirm "reality."

But the events of Denmark also
includes Hamlet's grief and his debate over taking the action necessary to revenge his
father's murder.  When he sees the First Player, in Act II, scene ii, present the
illusion of real grief -- "all his visage wann'd/Tears in his eyes" -- he chides himself
that he, with a "real" reason for grief can do
nothing.

Also, the speech delivered by the First Player is
the retelling of the death of King Priam and the grief-stricken response to this of his
queen, Hecuba.  This "illusion," again, contrasts the "reality" of the situation in
Denmark where, rather than rage in grief as Hecuba does, Queen Gertrude has married her
husband's murderer.

The players play a significant part in
demonstrating the theme of appearance versus reality in the play
Hamlet, contrasting the emotional highs and overt displays of grief
enacted by the player with the inaction of Hamlet and the un-widow-like behaviour of
Gertrude with the story of Hecuba.  It is, however, "The Mousetrap" that presents the
most important connection to this theme, wherein Hamlet uses "appearance" to prove
"reality."