Please provide arguments and quotes explaining why Romeo was correct in killing Tybalt and why he shouldn't be punished. Also provide arguments...

It's difficult to make the case that any of the characters in Shakespeare's Romeo and Juliet deserve to die. The two who die first, Mercutio and Tybalt, may deserve it the most because of their aggressive attitudes about the feud. It is obviously a matter of interpretation as to whether Tybalt warrants killing by Romeo. Arguments can be made on both sides. 

Romeo should go unpunished because Tybalt is antagonistic and stubborn. When he enters...

It's difficult to make the case that any of the characters in Shakespeare's Romeo and Juliet deserve to die. The two who die first, Mercutio and Tybalt, may deserve it the most because of their aggressive attitudes about the feud. It is obviously a matter of interpretation as to whether Tybalt warrants killing by Romeo. Arguments can be made on both sides. 

Romeo should go unpunished because Tybalt is antagonistic and stubborn. When he enters Act I, Scene 1, he threatens to kill Benvolio for no good reason other than Benvolio's name. Benvolio is simply trying to keep the servants of the two families from fighting in the streets of Verona. Tybalt challenges Benvolio. He says,

What, drawn and talk of peace? I hate the word
As I hate hell, all Montagues, and thee.
Have at thee, coward!

Later at Lord Capulet's party he overhears Romeo and flies into a rage, fully intending on disrupting the party by fighting his perceived foe, even though Romeo is there with peaceful intent. In Act I, Scene 5, he says,

Fetch me my rapier, boy.
What, dares the slave
Come hither covered with an antic face
To fleer and scorn at our solemnity?
Now, by the stock and honor of my kin,
To strike him dead I hold it not a sin.

He feels it is his right to kill any Montague he comes across. After being dissuaded from fighting at the party by Lord Capulet he sends a letter to Romeo challenging him. 

In Act III, Scene 1, Tybalt is again the troublemaker as he comes upon Mercutio and Benvolio on the streets. His challenge is for Romeo but he is also willing to clash with Mercutio when the time is right. He says,

You shall find me apt enough to that, sir, an
you will give me occasion.

When Romeo shows up, Tybalt is not satisfied when the son of Montague backs down. Romeo, unknown to anyone else in the scene, has just married Juliet and won't engage Tybalt. Tybalt, not satisfied, continues to press for a fight. He says,

Boy, this shall not excuse the injuries
That thou hast done me. Therefore turn and draw.

After Mercutio is killed, Tybalt initially runs away but, because of his belligerent nature, returns to the scene and is perfectly willing to fight Romeo, who feels it is his duty to avenge his friend. Tybalt says,

Thou wretched boy that didst consort him here
Shalt with him hence.

Tybalt could have acted differently. Unfortunately, his anger and love of fighting got the best of him. Romeo is within his rights to kill him and should not be punished.

An opposing argument could also be made:

Tybalt doesn't deserve to die because he never intended to kill Mercutio. The two are engaged in sword play when Romeo tries to stop them, causing Mercutio to be stabbed under Romeo's arm. He bears no ill will toward Mercutio and tries to avoid fighting Mercutio when Romeo shows up. He says, in Act III, Scene 1, 

Well, peace be with you, sir. Here comes my man.

As he is dying Mercutio confirms that it was Romeo's fault that Tybalt stabbed him. He says, 

Why the devil came you between us?

I was hurt under your arm.

Mercutio blames both Tybalt and Romeo for his death as he utters the famous lines cursing the Montagues and Capulets:

A plague o’ both your houses!

Zounds, a dog, a rat, a mouse, a cat, to scratch a man to death!

Finally, it may be said that although Tybalt is quarrelsome he also has redeeming qualities and is loved by the Capulet family. These feelings are expressed by the Nurse in Act III, Scene 2. She says,

O Tybalt, Tybalt, the best friend I had!
O courteous Tybalt, honest gentleman,
That ever I should live to see thee dead!

The banishment of Romeo is totally fair. In fact, he is getting off easy because the Prince had said in Act I that the death penalty would be levied for fighting in the street after the brawl which opens the play. Romeo didn't need to killed Tybalt and should be relieved at the Prince's verdict.