How many kids did William Shakespeare have and what happened to his son?

In 1582, Shakespeare married a woman eight years his senior, Anne Hathaway. The couple first had a daughter named Susanna, then twins - Judith and Hamnet. The circumstances of Hamnet's death and the effect of this upon Shakepeare remain unclear, but we know that in the summer of 1596 the playwright and actor received news that his eleven-year-old son was terribly ill:

Whether in London or on tour with his company he would at best have only been able to receive news intermittently from his family in Stratford, but at some point in the summer he presumably learned that Hamnet's condition had worsened and that it was necessary to drop everything and hurry home. By the time the father reached Stratford the boy—whom, apart from brief visits, Shakespeare had in effect abandoned in his infancy—may already have died. On August 11, 1596, Hamnet was buried at Holy Trinity Church: the clerk duly noted in the burial register, "Hamnet filius William Shakspere."

Some critics interpose and say that the probabilities are high that the child had contracted the bulbonic plague, highly contagious and very widespread at the time. This remains pure speculation, but Shakespeare was not outwardly demonstrative in his mourning. Instead, he plunged into his work again, writing some of his merrier works, perhaps as a kind of defense mechanism to spite fate:

In the four years following Hamnet's death, the playwright, as many have pointed out, wrote some of his sunniest comedies: The Merry Wives of Windsor, Much Ado About Nothing, As You Like It. This fact is, for some, decisive evidence that the father's grief must at most have been brief. But the plays of these years were by no means uniformly cheerful, and at moments they seem to reflect an experience of deep personal loss. In King John, probably written in 1596 just after the boy was laid to rest, Shakespeare depicted a mother so frantic at the loss of her son that she is driven to thoughts of suicide:

Grief fills the room up of my absent child,
Lies in his bed, walks up and down with me,
Puts on his pretty looks, repeats his words,
Remembers me of all his gracious parts,
Stuffs out his vacant garments with his form.

Based on such sketchy evidence, we can only speculate on poor Hamnet's fate and how this event influenced Shakespeare's writings and personal life thereafter. Four years later, though, Shakespeare completed his tragedy "Hamlet," based on a story already anchored in legend. Was the uncanny similarity in name a mere coincidence with Shakespeare's "...to be or not to be?", or was this Shakespeare's final expression of his grief over the loss of his son?