Do the King and Queen believe that Hamlet is mad? Why or why not?

As of Act 3, Scene 1, the King does not believe Hamlet is mad. After Claudius and Polonius have been eavesdropping on Hamlet's quarrel with Ophelia, the King says:

Love? His affections do not that way tend;
Nor what he spake, though it lack'd form a little,
Was not like madness.There's something in his soul
O'er which his melancholy sits on brood;
And I do doubt the hatch and the disclose
Will be some danger;

Hamlet's behavior at the play-within-a-play in Act 3, Scene 2, causes the Queen to think her son must be mad. And when he comes to her chamber in Act 3, Scene 4, she is sure he is mad because she thinks he intends to murder her. She starts calling for the guards, and Polonius, who is hiding behind the arras and can't see anything, starts echoing her cries for help. This leads to Polonius' death.

But then in the tempestuous scene with his mother, Hamlet convinces her that he is 

not in madness but mad in craft

However, he persuades her not to reveal this truth to Claudius. When she goes to see her husband in Act 4, Scene 1, she tells him that Hamlet is

Mad as the sea and wind, when both contend
Which is the mightier. In his lawless fit,
Behind the arras hearing something stir,
Whips out his rapier, cries, "A rat, a ra t!"
And in this brainish apprehension kills
The unseen good old man.

Now Claudius begins to believe that his stepson must be mad, although, ironically, Gertrude no longer believes it but is lying when she says, "Mad as the sea and wind, etc." The Queen will not believe her son is mad for the rest of the play. The King will believe it after Hamlet puts on a show of madness for him in Act 4, Scene 3. Claudius says:

Alas, alas!

It is important to note that the King is now concerned about his own safety. If Hamlet can kill Polonius in a fit of madness, what is to keep him from doing the same thing to Claudius? The King decides to pack Hamlet off to England immediately, and at the end of Act 4, Scene 3, he reveals in a soliloquy that the letters he is sending with Hamlet, Rosencrantz and Guildenstern call for

The present death of Hamlet.

So the Queen believes Hamlet is mad and then stops believing it after the big meeting with her son in her chamber. But she tells the King her son is mad to protect him from punishment for murdering Polonius. The King started believing Hamlet is mad because of what his wife tells him, and he is convinced of Hamlet's madness because of the act his stepson puts on when he appears before him, talking about how Polonius is in conference with worms and other bizarre things.

In truth, it would appear, Hamlet was never mad from the beginning to the end of the play. After his meeting with the Ghost in Act 1, Scene 5, he swears his friends to secrecy and tells them he 

perchance hereafter shall think meet
To put an antic disposition on...

He has been spied up relentlessly since returning from Wittenberg, and he foresees the possibility that the spying will be stepped up in the future because he knows he is a changed man after what he has been through with a ghost. Hamlet keeps everybody guessing. He is especially afraid of Claudius, who is very suspicious of him already. The play is largely charged by a battle of wits between these two clever men.