What crimes is Kit charged with in chapter 19 of The Witch of Blackbird Pond?

The broad overall charge brought against Kit in chapter 19 of The Witch of Blackbird Pond is witchcraft and consorting with the devil. 

"Katherine Tyler, thou art here accused that not having the fear of God before thine eyes thou hast had familiarity with Satan the grand enemy of God and man, and that by his instigation and help thou hast in a preternatural way afflicted and done harm to the bodies and estates of sundry of His Majesty's subjects, in the third year of His Majesty's reign, for which by the law of God and the law of the Colony thou deservest to die."

Specifically, Kit is accused of working with the devil to hurt the members of the town. Kit is accused of consorting with Hannah Tupper, whom the townsfolk believe is a witch. Kit supposedly learned witchcraft from Hannah. To make matters worse for Kit, the charges specify that Kit has been using her devil witchcraft powers to hurt the innocent children of the town. She supposedly makes them sick and die. 

"Mistress Tyler, you are accused by Adam Gruff with the following actions. Firstly that you were the familiar friend and companion of the Widow Hannah Tupper of Blackbird Pond, an alleged witch who has within the past week disappeared in a suspicious manner. Such friendship is a lawful test of guilt, inasmuch as it is well known that witchcraft is an art that may be learned and conveyed from one person to another, and that it has often fallen out that a witch, upon dying, leaveth some heir to her witchcraft. "Secondly, that you are guilty of actions and works which infer a court with the devil, which have caused illness and death to fall upon many innocent children in this town."

Kit is then asked if she knows Hannah Tupper.  Yes, she knows Hannah.  Then Kit is asked if she has seen Hannah's cat.  Yes to that one too. The next question asked of Kit is whether or not she has done some "enchantments" with Hannah.  Kit replies no. 

Kit proves her innocence with the help of Nat. Together they prove that Kit is not a witch because she helped Prudence learn to read the Bible. 

What happens to Kit during her witch trial in "The Witch of Blackbird Pond"?

Kit is brought before the Magistrate for an inquest to determine if she should be brought to trial in Hartford.  The proceedings begin with a reading of the charges against Kit, after which she is questioned as to her relationship with Hannah Tupper.  After it is established that Kit had indeed befriended the old woman in defiance of her Uncle's orders, various citizens come forth with outrageous complaints of witchcraft purportedly executed by Hannah and Kit.  Outraged, Matthew Wood offers to vouch for his niece's character, but his credibility is undermined because it has already been shown that she has deceived him by continuing her visits to Hannah.  Goodman Cruff then steps forward with what he thinks is damaging evidence, a copybook with his daughter's name written repeatedly on it, as if for a spell.  When asked to explain the copybook, Kit refuses, not wanting to involve the child in the proceedings.

Because of her refusal to cooperate, Kit is about to be remanded to trial, when Nat Eaton appears with Prudence Cruff.  Prudence tells the court that it is her writing in the copybook, and that Kit has also taught her to read.  To the astonishment of her father, who has believed his wife's assertions that the child is witless, Prudence proves that she can write her name, and reads from the Bible before the assembly.  The court finds "no evidence of witchcraft", and the charges against Kit are dropped (Chapter 19).

In "The Witch of Blackbird Pond," why was Kit arrested? What were the underlying causes?

This question accurately asks about multiple underlying causes about why Kit is arrested and put on trial for witchcraft. The first cause is that Kit is an outsider. She hasn't grown up in the colony, and she hasn't grown up in the Puritan way of life. All that this does is make her different. She has a different viewpoint on all kinds of things. From her view of work to how she dresses, Kit doesn't fit the mold that exists around her. Second, Kit is friends with Hannah Tupper. Hannah is also an oddity in the area, and the rumor of witchcraft is always bubbling under the surface about her.

Quakers cause trouble wherever they go. They speak out against our faith. Of course, we don’t torment them here in Connecticut. In Boston I’ve heard they even hanged some Quakers. This Hannah Tupper and her husband was branded and driven out of Massachusetts. They were thankful enough just to be let alone here in Wethersfield.

Third, the mysterious illness strikes the town. The Puritans understand that people get sick, but this illness is beyond the normal cold; therefore, the superstitious attitude of the people begins to suspect witchcraft. Hannah Tupper is a convenient scapegoat, but Kit manages to help her escape. The result is that with Hannah missing, Kit is the next best scapegoat.

In "The Witch of Blackbird Pond," why was Kit arrested? What were the underlying causes?

Kit was arrested because she became a threat to the colony. She made the decision to stand and face the hypocrisy she witnessed in the colony, and for this she paid a price. It is important to remember that Puritan America had little tolerance for the individual. The idea that a woman should think for herself or own her own conscience was enough for her to be deemed a witch. Historically, this was a very convenient way for the Puritan elder hierarchy (all of whom were men) to keep women who were strong of spirit in check. Kit's character defied that hierarchy. She concluded that her loyalty belonged to her beliefs, which included standing up for her friend Hannah. Hannah was the only person who truly accepted Kit on her own terms, the only one with whom she could be true to herself without reservation. Nat also plays a significant role in the novel in that his character confirms the hypocrisy of the colony and realization that Kit's kindness, loyalty to those she loved, and principle of dignity was worth the consequences she faced.

What is Kit's punishment in The Witch of Blackbird Pond?

I believe that this question is asking about events that begin in chapter 9. This chapter begins with Kit and Mercy starting to teach school at the Wood's home. Kit is not nearly as patient as Mercy, but she is good at teaching the older kids to read through her usage of "tricks" and rhymes. The two teach eleven total children, from ages four to seven.

At one point in the teaching day, Kit decides to tell the story of the Good Samaritan. Kit is fairly adept at telling stories, but she decides that in order to make the story more real, tangible, and fun for the kids, the kids will act out the various parts of the story. Mercy is not fully on board with the idea. Playacting in a Puritan school is not something that is done.

Unfortunately, the entire idea goes from bad to worse. The boys start goofing off instead of taking it seriously, and that is the moment that Mr. Eleazer Kimberly and Reverend John Woodbridge enter the classroom. Kit takes the blame, and the men are very disappointed.

"Play-acting! And with the Bible."

Reverend Woodbridge stared incredulously at Mercy. "What could you have been thinking of, Mercy, to allow such a thing?"

...

"I am shocked and disappointed," he said sternly. "I had heard such excellent reports of your school."

The punishment is quick and severe. School is dismissed for the day and the following day. Additionally, Kit is permanently dismissed as the teacher, and the two men will "consider seriously" whether or not Mercy should be fired as well.