Who is spying, and on whom?
Claudius and Gertrude set Rosencrantz and Guildenstern to spy on Hamlet. The new King has summoned Hamlet’s schoolmates in order for them to spy on the Prince and report back to Claudius, recounting every move Hamlet makes.
Where is the audience (and the characters in the play) given second-hand accounts of things that happen in the play (think Ophelia, Voltimand, Cornelius)?
The audience and characters in the play are given second-hand accounts of things that happened when Polonius and Hamlet are talking, then multiple actors walk in which leads to Hamlet talking about the play he wants to put on and gives second-hand accounts of everything that he has experienced.
What are the liabilities of second hand information?
It's not always told in the right context, and it can easily be morphed into something completely different, even with a specific tone that the second hand information is told in.
What is the effect of the characters not addressing each other directly in order to understand each other? Why spy?
It gives a sense of mystery and risk by spying. If they get caught, then they can't be trusted anymore, but since they do it without getting caught, it gives the reader information that Hamlet wouldn't want us to know
What does Shakespeare seem to be implying by having so much spying and indirect action in the play?
This would indicate a lack of trust they have for each other; they do not trust the words and actions seen by others, so spying appears to be the only way to determine the true intentions they have for each other and how it will affect them.