Is Gertrude protecting Hamlet when she says he is mad (4.1.7), or does she believe that he is mad? If she believes he is mad, does it follow that she no longer feels ashamed and guilty? Explain.

It would be difficult for Gertrude, or for any mother under the circumstances, not to believe that her son is mad after the way he has behaved. But, of course, she's nevertheless trying to protect him, to tell the others that, in spite of his violently disruptive behavior, Hamlet, given his mental condition, should not be harmed.
Gertrude does not come off as one of the more intelligent or perceptive of Shakespeare's characters. From the start, she seems not to grasp the extent to which Hamlet is traumatized by the death of his father. Nor does she grasp the hostility he feels toward his new stepfather, Claudius. Even if she is more sensitive than she seems to the real reason for Hamlet's dysfunctional manner, she's in denial about it. She wishes above all to maintain the status quo of her new marriage. To recognize why Hamlet acts as he does would destroy that marriage. At the same time, it's probable that she does feel guilt and shame—again, this is almost instinctive for any mother to do when her child is so obviously pathologically disturbed, as Hamlet is.
When Hamlet confronts her in act 3, scene 4, Gertrude is stunned into recognizing how warped Hamlet's personality has become. When she tells him, "Thou hast cleft my heart in twain," it is not only because of his hateful posture toward her but also, arguably, because she feels responsible for the situation, for what Hamlet has become and for the mayhem he's causing. At Ophelia's funeral, before Hamlet bursts onto the scene, Gertrude sadly remarks that she had wished Ophelia for his wife. This remark, perhaps more than anything else, reveals the plans Gertrude had entertained for her son to have a normal life, to become a functional person.
One could, of course, argue that even if she did not believe Hamlet to be insane, she would still try to make the others think he was in order to protect him. But, especially after he has killed Polonius without showing the slightest regret, and, moreover, after he has hallucinated a ghost (because Gertrude herself does not see the ghost, she naturally would believe Hamlet is hallucinating), why would Gertrude not think him insane? Even if Hamlet were shamming madness as part of an elaborate hoax for his purpose of revenge, any person would, in my view, have to be insane first in order to perpetrate such a ruse.

Gertrude certainly appears to think that Hamlet is insane. She tells Claudius frankly in act 4, scene 1, that he is as mad as the sea and the wind in a storm. Whether Gertrude really does think that Hamlet is mad is a matter of debate, but there seems to be little doubt that by telling Claudius that Hamlet is mad, she is putting her faith in her new husband. Now, whatever Hamlet's true psychological state, she has effectively thrown in her lot with Claudius. Her greater loyalty appears to be to him and not to her son. Gertrude's description of Hamlet's mental state validates Claudius in his plan to send Hamlet to England to have him assassinated. Of course, this is not intentional on Gertrude's part; there is nothing to suggest that she wants any harm to come to her son. However, at the very least, her growing estrangement from Hamlet means that she is no longer fully able to protect him, even if she wants to.