It's impossible precisely to say. There is a case to be made in the text for either possibility.
To argue that she did know about Old Hamlet's murder, people turn to the evidence that her marriage to Claudius, she says, is "o'erhasty" - as if they should have waited to allow suspicion to subside? You can also play her reactions in the play scene as deeply uncomfortable ("the lady doth protest too much").
To argue that she didn't know, productions (such as the recent staging by Sir Trevor Nunn at London's Old Vic Theatre) simply have to play Gertrude as not very bright, and not very knowing. So Gertrude is horrified at the play, but doesn't put two and two together, and - potentially - only realises during the closet scene (Act 3, Scene 4), which then (in some productions) leads to her deliberately drinking the poison in the final scene.
Even the ghost is ambiguous - is she to go to heaven, or go the thorny way:
Taint not thy mind, nor let thy soul contrive
Against thy mother aught. Leave her to heaven,
And to those thorns that in her bosom lodge
To prick and sting her.
You can do it both ways. If you wanted a definite answer, it depends on how you read the closet scene, and Gertrude's behaviour after it. When Hamlet calls Claudius "a murderer and a villain" - does she understand what he means, specifically? Does she stop the "bloat king" Claudius from tempting her to bed after that scene? Shakespeare doesn't specify. It's up to your own reading of the text to make an argument for what Gertrude really knew.
I agree that it is difficult to tell from the dialogue of the accusation. "To kill a King?!" The actor might have great latitude in how to act this line. The queen might be fretting over the discovery of her knowledge, or she might be genuinely shocked at the notion. Gertrude is an intriguing character. Lady MacBeth's schemes are clear and blood curdling. Gertrude may be played as cool and calculating, or she may be an empty headed pawn. I believe Shakespeare was allowing for this to be a highly interpreted character and the participation of both the actor and the audience fills that void. This lacuna invites this very question, among others.
No. There is nothing in the text to support that Gertrude knew of Claudius' deed. But, Hamlet doesn't know that. The first part of the play within the play, i.e., The Mousetrap, is testing Gertrude's involvement with King Hamlet's death. She of course passes the test. Later in her closet when confronted by Hamlet with the accusation that the King was murdered, she shows no indication of guilt. Nor does the play provide a reason for Gertrude to need such knowledge.
I will disagree with the prior referenced post. This is not an unsolved crime. There is no real Gertrude for whom we may speculate her true involvement. If Shakespeare wanted Gertrude to know of the murder then it would be in the play. Though the question is at issue within the play it is not left unresolved by the end of the play. Clearly, Gertrude knows nothing of Claudius' guilt.