In The Witch of Blackbird Pond, what was Nat's punishment?

In The Witch of Blackbird Pond, Nat is, in fact, actually punished in Chapter 16, well before the aforementioned threatened punishment (Chapter 19). He is one of "a rowdy band of rivermen from a trading ship" that comes running into Whethersfield on All Hallows Eve to vandalize the house of the man Kit has been seeing (William Ashby) by placing "lanterns . . . made out of pumpkin heads, with candles inside, and unholy faces cut in the sides to show the light," which Kit's own uncle declares "the devil's invention" with much solemn disapproval. Furthermore, the so-called "rowdy band" stole the pumpkins out of a field to turn them into the jack-o-lanterns.

For those very crimes, on "Thursday Lecture day, the day of public punishment" later that week, Kit slips out and goes alone to the Meeting House, where she finds three men from the Dolphin, Nat's father's ship, in the stocks. Nat, of course, is one of the three, apparently unable to stay out of trouble, even indirectly, at least where Kit or the law might be concerned. The rest of the punishment, aside from time in the stocks (which brought along with it the public derision, scorn, and even abuse of passersby), includes "a fine of forty shillings each" and the reproof to never set foot in Whethersfield again, lest they be whipped (see the threat mentioned in Chapter 19).

You might find it useful to refer to the specific part of the novel that your question pertains to. This section of the novel comes in Chapter 19, when Nat so gallantly and bravely returns to rescue Kit from being tried as a witch. He brings Prudence with him who shows everyone how Kit taught her how to read. Of course, it is when Goodwife Cruff has her revenge against Kit spoilt that she turns it onto Nat:

"That man!" she shrilled. "Isn't he the seaman? The one who was banished for setting fire to houses? Thirty lashes they promised him if he showed his face here again!"

Although Nat manages to slip away before they capture him, his willingness to show up and defend Kit to see justice done against the ignorance of the community shows what kind of man he is and demonstrates both his love of Kit but also his love of justice.

In The Witch of Blackbird Pond, why was Nat put in the stocks?

Nat was put in the stocks because he was one of the sailors who put Jack-o-lanterns in the home of William Ashby.  It was just a joke, but Ashby was probably chosen because Nat was upset that he was going to marry Kit.  Ashby did not see the incident as a joke.  He saw it as satanic.  This is hard for us to understand, but All Hallows Eve was considered by many Puritans to be a satanic holiday and anything associated with it to be satanic.  Some Christians today are still offended by jack-o-lanterns and other halloween celebrations. 

On All Saints' Day, which the Puritans regard as a pagan feast, according to Matthew, Nat places lanterns in the windows of William Ashby's house. The windows do not yet have panes, and Nat and two of his friends place lanterns made out of pumpkin heads with candles inside into the windows. The outside of the pumpkins are carved into what are considered unholy faces. Uncle Matthew and other Puritans regard these carved faces as "blasphemous" (165), or the work of the Devil. Nat and two other men from the Dolphin sit in the stocks, and they show no signs of being sorry for what they did. When Kit sees Nat in the stocks, she feels sorry for him and asks if she can do anything to help him, but he denies that he is uncomfortable.

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.

What are two quotes about Nat being punished in the stocks in Witch of Blackbird Pond?

About nine paragraphs into Chapter 16, Matthew Wood tells his family that

"the three ringleaders (of the prank) are cooling their heels now in his shed, and on Lecture Day they will sit for all to see in the town stocks".

As it is described about two pages later, Kit feels compelled to go down to the stocks on Lecture Day, even though "it was no place for a girl alone".  There, confined in the wooden restraints, she sees

"all three Dolphin men...Nat and the redheaded seaman who had painted the Dolphin's figurehead...cheerfully exchanging insults with a cluster of young bound boys...the two culprits holding their own in an unchastened manner...even as (Kit) watched, an apple core sailed through the air and bounced off Nat's forehead".

When Nat's attention is directed towards Kit, he

"twist(s) his head the inch or so that the boards (allow) him and stare(s) at her without the slightest recognition".

When he realizes who she is, he exasperatedly tells her to "get away from this place".

A few paragraphs later, Kit arrives at the Meeting House where she stops to read the posted notice which says,

"That for stealing pumpkins from a field, and for kindling a fire in a dwelling they three shall be seated in the stocks from one hour before the Lecture till one hour after" (Chapter 16).

What action led to Nat's punishment and fine in The Witch of Blackbird Pond?

Nat set up jack-o-lanterns at William Ashby's house. The pumpkins caught fire and eventually burnt down William Ashby's entire house.

Someone had given Nat a note that expressed the fine would get paid for Nat and Nat would additionally receive 40 shillings. This note was completely anonymous making the reader question which other character could have possibly encouraged this behavior.

Ironically, this event that caused Nat great trouble occurs on All Hallow's Eve. It also appears that Nat enlisted the help of some of his sailor friends. These two helpers also received 40 shillings and a punishment of a period of time in the stocks.

How are Nat and the others punished for the prank in The Witch of Blackbird Pond?

Nat and Kit's relationship has a somewhat rocky start. First, Nat is embarrassed and angered by Kit when she jumps into the water to save Prudence's doll. He splashes in after her but feels silly instead of heroic when he learns that she is in no danger of drowning and is upset that he has drenched his only set of clothes. 

However, later Nat learns that Kit has been helping Hannah Tupper, an old woman who lives near Blackbird Pond. Since Nat is a good friend of Hannah's, he is touched that Kit has also been taking care of her. The two grow closer while doing chores for Hannah and Nat begins to fall for Kit.

With this in mind, it is obvious that Nat is jealous when he learns that William Ashby intends to marry Kit and is constructing a large new house where they will live. Nat and some friends from The Dolphin put jack-o-lanterns in the windows of William's new house and are caught and punished by being put in the stocks.

While in the stocks, the townspeople taunt the wrongdoers and one throws an apple at Nat's head. Kit is visibly upset by this, but Nat tells her it was worth it to see the look on William's face.