Who was the Thane of Cawdor before Macbeth?

The Thane of Cawdor is not named in Macbeth. But, in Act I, Scene 2, the Thane of Cawdor is labeled a Scottish traitor by the Thane of Ross when he returns to camp. Further, Ross reports that the Norwegians, who have been in "terrible numbers," fought with the king's troops. In fact, the king of Norway himself did battle with the Scots. His soldiers were aided by the Thane of Cawdor, called "the disloyal traitor" by Ross. It has been a "dismal conflict," one that threatens Scotland, in which they all engaged. But, after the mighty Macbeth reached the battlefield, the Scots defeated Norway.

Hearing this report, Scotland's king, Duncan, tells Ross that he wants Macbeth awarded the title Thane of Cawdor because of his bravery and skill. Furthermore, King Duncan orders that the traitorous Thane of Cawdor be put to death and his title awarded to Macbeth for his bravery. "What he hath lost, noble Macbeth hath won" (1.2.68).

How does Macbeth become Thane of Cawdor in Macbeth?

The Thane of Cawdor is a traitor and Macbeth is promoted to his title.

Macbeth distinguishes himself in battle, and then hears the prophecies of the three witches.  One of these is that he will be Thane of Cawdor.  The other two prophecies are that he will be king, and that Banqo’s sons will be king.  When Macbeth appears at Duncan’s castle, Ross addresses him with his new title.  That is how he finds out he has been promoted due to his recent battle heroics and the downfall of the person who last had that title.

MACBETH
The thane of Cawdor lives: why do you dress me
In borrow'd robes?
ANGUS
Who was the thane lives yet;
But under heavy judgment bears that life
Which he deserves to lose. … (Act 1, Scene 3)

Macbeth doesn’t seem to be too upset that the current Thane of Cawdor has confessed to being a traitor.  He is more interested in the fact that one of the witches’ prophecies came true.  If one came true, can the others be far behind?

Ever since the witches planted this seed in his head, Macbeth cannot be satisfied.  He wants to be king, because he has been told that he will be king.  When Duncan announces that his son will be his successor, which in most circles would be expected, Macbeth is upset.  He wants all of the witches prophecies to come true, or at least those that relate to him.  He doesn't want Banquo's sons to be king!

Who becomes Thane of Cawdor after Macbeth is crowned king?

Nobody succeeds Macbeth as Thane of Cawdor within the context of the play. As Banquo says to him, in act 3, scene 1, "thou hast it now: king, Cawdor, Glamis, all." Macbeth, having become Thane of Glamis first, can simultaneously hold this title alongside that of Thane of Cawdor; having become king, he simply maintains all three positions, meaning that he is able to wield power over Glamis and Cawdor specifically, which in theory would enable him to be a more powerful king. In practice, this does not work out for Macbeth, of course. We would expect the titles of Thane of Cawdor and Thane of Glamis to be passed down from Macbeth to his sons, which Macbeth does not get the chance to do.

In a modern monarchy, it would be very unlikely for a king to maintain any previous lesser titles—for example, King George VI of Great Britain was previously the Duke of York, a title which he renounced upon becoming king. But it remains perfectly possible for a Prince to simultaneously hold a number of other titles: Prince William of Great Britain is also Duke of Cambridge, Earl of Strathearn, and Baron Carrickfergus. Because Macbeth takes place in a time when the king and thanes of Scotland held significant personal and political power, it would not make sense for him to renounce his thaneships, as that would give other people the opportunity to build their own powers. Had he continued as king, he could have awarded this thaneship to a loyal follower, as Duncan awarded it to Macbeth.

What happens to the former Thane of Cawdor in Macbeth?

Renowned Shakespeare scholar Harold Bloom calls Macbeth 

...an uncanny unity of setting, plot, and characters, fused together beyond comparison with any other play of Shakespeare's.

Certainly, Macbeth exhibits an excess of blood and is even referred to as "a tragedy of blood" in its murders and in the imaginations of such as Macbeth and Lady Macbeth. The first bloody murder is that of the former Thane of Cawdor in Act I. In Scene 2, the king, Duncan, and his forces have engaged in battle with the King of Norway and the traitor Macdonwald. Just then, a wounded captain arrives after having saved Malcolm, Duncan's son, from the battlefield. Malcolm, then, has the captain relate what has transpired in the field. The king, then, learns that brave Macbeth wrestled for a time with the traitorous Macdonwald

Disdaining fortune, with his brandished steel,
Which smoked with bloody execution,
Like valor's minion carved out his passage
Till he faced the slave:...
Till he unseamed him from the nave to th'chops,
And fixed his head upon our battlements. (1.2.17-22)

Hearing of this bloody victory, Duncan declares, "O valiant cousin! Worthy gentleman!" And, when he learns that Macbeth and Banquo were not the least dismayed by the fresh attack from Norway, King Duncan then meets with Ross and Angus who report that the Thane of Cawdor "began a dismal conflict" until Macbeth defeated the troops. The king, then, tells Ross,

No more that Thane of Cawdor shall deceive
Our bosom interest: go pronounce his present death,
And with his former title greet Macbeth. (1.2.64-66)

In the meantime, Macbeth and Banquo cross a heath near Forres and encounter three witches, who are stirring a cauldron. When Macbeth addresses them, the first witch greets him, "All hail, Macbeth! Hail to thee, Thane of Cawdor!" (1.3.50). Later on, Macbeth encounters Ross and learns that he has, in fact, been made Thane of Cawdor. He wonders then, too, that having correctly predicted his earning the title of Thane of Cawdor, if the witches were also correct in their predicting that he also would be king.