Is Hamlet surprised when the ghost asks him to revenge his father's murder? Is he surprised when he knows who the murderer is?

Yes, on both accounts.  In Shakespeare's
Hamlet, Hamlet is deeply troubled by his father's unexpected death,
his mother's hasty remarriage to his uncle (seen as incestuous), and, presumably, the
loss of the throne.  But there is no evidence that Hamlet suspects his father was
murdered. 

When the Ghost first mentions revenge, Hamlet,
known for his profound speeches,
responds:

What?  (Act
1.5.8)

When the Ghost
continues:

readability="5">

Revenge his foul and most unnatural
murder.

Hamlet, still
shocked, responds:

Murder! 
(Act 1.5.26)

When the Ghost
repeats himself, Hamlet pleads for an explanation:

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Haste me to know't, that I, with wings as
swift

As meditation or the thoughts of
love,

May sweep to my revenge.  (Act
1.5.29-31)

And when the Ghost
reveals the identity of the murderer to be Claudius, Hamlet seems genuinely
surprised:

O my prophetic
soul!

My
uncle!

We have to assume that
if not for the Ghost, Claudius would have gotten away with murder.  Hamlet had no
idea.