[Editors are only permitted to answer one question per posting.]
In Shakespeare's Hamlet, I sympathize with Gertrude. With her husband dead, her future is very much in jeopardy. I think she is motivated to marry in order to survive. When Claudius marries Gertrude, it guarantees that she will have a home—and Hamlet as well. Women had no rights when Shakespeare wrote this play, regardless of their social standing—they were ruled by the decisions of men. If Gertrude had not married Claudius, can we know for certain that he would not have put her aside (which might well have meant she would have had to join a convent to survive)?
According to Elizabethan society, Gertrude commits incest in marrying her brother-in-law because they believed that when a man and woman married, they were one person. Therefore, when Old Hamlet dies, part of him remained within Gertrude, and marrying Claudius meant that the brothers were, in essence, sleeping together. Her surprise later over Hamlet's accusations makes me think she may also have succumbed to Claudius' wooing.
Hamlet is angry that Gertrude seemed to get over Old Hamlet's death so quickly, and then enters an incestuous marriage. He is unforgiving of this, even before he discovers that Claudius murdered his father.
Later, after Hamlet accidentally kills Polonius, Hamlet tells Gertrude that Claudius is a murderer; she seems completely unnerved.
...A bloody deed. Almost as bad, good mother,
As kill a king, and marry with his brother.
As kill a king?
Ay, lady, it was my word. (III.iv.31-34)
Not only is Gertrude seemingly amazed by her son's accusation, she seems genuinely confused by his anger:
What have I done, that thou darest wag thy tongue
In noise so rude against me? (43-44)
Hamlet does not deliver his displeasure lightly, but goes at his mother with a force that makes her look deeply within herself:
Look here upon this picture, and on this,
The counterfeit presentment of two brothers.
See what a grace was seated on this brow… (59-61)
...A combination and a form indeed
Where every god did seem to set his seal
To give the world assurance of a man.
This was your husband. Look you now what follows.
Here is your husband, like a mildew'd ear
Blasting his wholesome brother. Have you eyes? (66-71)
… what judgment
Would step from this to this?… (76-77)
O shame! where is thy blush? (88)
The Queen is mortified by what she sees when she looks at her situation through Hamlet's eyes, and she begs him to stop:
O Hamlet, speak no more!
Thou turn'st mine eyes into my very soul,
And there I see such black and grained spots
As will not leave their tinct... (95-98)
O, speak to me no more!
These words like daggers enter in mine ears.
No more, sweet Hamlet! (103-105)
When the Ghost tells Hamlet to be gentle with his mother, Hamlet turns compassionate eyes toward her, asking if she is all right; she, in turn, worries for him (as he's been talking to the air, so it seems). Gertrude loves her son. (Her loving nature is seen in how she suffers over Ophelia's death.)
Gertrude is motivated at first to survive. Her shame over Hamlet's observations of her character mortifies her. If she were guilty of deceit, she would be unmoved by Hamlet's words. She is astounded when Hamlet infers that Claudius killed Old Hamlet. She cares for herself and worries for Hamlet's future. She loves her son, and loved Old Hamlet. She has a good heart.