In Act 1 scene 3, lines 60-141 and scene 4, lines 1-40 of Hamlet. What symbols are identified and occur throughout the play?

The language in Hamlet is so rich
that any section of the text would provide many of examples of the literary elements
that  you named.  Since your question focused on symbols, I'll confine my response to
these.  My line numbers are slightly different from yours, but the passage you mentioned
includes Polonius's advice to Laertes and to Ophelia, followed by Hamlet's conversation
with Horatio.  One of the major symbols mentioned in this selection is that of traps.
 Polonius warns Ophelia that Hamlet's words are

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springes to catch
woodcocks!

This imagery
becomes quite symbolic as many of the characters in the play are trapped and destroyed
by their own choices.  We see that Gertrude is trapped in her marriage to Claudius;
Claudius feels that his soul is trapped because of the murder of his brother; Hamlet
feels trapped by his mission to avenge his father's death; Laertes himself is caught in
his own trap, killed by the poisoned sword he used on Hamlet, Ophelia is trapped by her
allegiance to her father.

Another symbol mentioned in this
passage is Hamlet's observation:  some men have a "vicious mole of nature" or a "dram of
evil" that corrupts the whole character of the man.  Hamlet here is discussing Claudius'
habit of drinking and partying.  But this mention of a small rotten defect that corrupts
the whole becomes symbolic of many of the characters as well as the Danish court.  A
small imperfection can grow to have a destructive defect.  This reference ties in very
well with Marcellus's famous line

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Something is rotten in
Denmark.

We see that Hamlet's
words can apply to Claudius's murder of his brother, an act that has far-reaching
consequences wreaking havoc on the entire Danish court.