In The Witch of Blackbird Pond, why does Nat get punished in the stocks?
Unfortunately I had to edit your question down as you asked more than one question. Enotes only allows you to ask one question.
Nat is placed in the stocks as a form of punishment, because he and some other sailors from his ship "illuminated" the house of William Ashby. Matthew Wood explains the meaning of this term:
"They put lanterns in the window frames that are waiting fro the new panes. Lanterns made out of pumpkin heads, with candles inside, and unholy faces cut in the sides to show the light."
Although Judith and Kit are able to recognise this for the harmless prank that it was, according to Matthew, as to wider Puritan society, they are regarded as "the devil's invention" and "outrageous blasphemy." However, key to realise is that Nat does this after finding out about the supposed engagement of Kit and William Ashby. Clearly this reveals some jealousy on his part, and some form of deep affection for Kit, just as when Kit approaches Nat in the stocks she reveals her feelings for him.