Is "Rapture" or "Caught up alive" in contradiction to God's words "Dust to dust"? By the sweat of your brow you will eat your food until you return to
  the ground, since from it you were taken; for dust you are and to dust
  you will return. (Genesis 3:19, NIV)
  
  Just as people are destined to die once, and after that to face
  judgment. (Hebrews 9:27, NIV)


From these verses, we can deduce that "All must die once (or at least once) and return to dust".


  For the Lord himself will come down from heaven, with a loud command,
  with the voice of the archangel and with the trumpet call of God, and
  the dead in Christ will rise first. After that, we who are still
  alive and are left will be caught up together with them in the clouds
  to meet the Lord in the air. And so we will be with the Lord forever.
  (1 Thessalonians 4:16-17)
  
  Listen, I tell you a mystery: We will not all sleep, but we will all
  be changed— in a flash, in the twinkling of an eye, at the last
  trumpet. For the trumpet will sound, the dead will be raised
  imperishable, and we will be changed. For the perishable must clothe
  itself with the imperishable, and the mortal with immortality. (1
  Corinthians 15:51-53, NIV)


These verses are very important basis for Christian eschatology belief called Rapture. As per these verses, on the the day of Rapture, first, the dead who belonged to Christ will be resurrected and they will be transformed, second, those who are alive and believe in Christ will be caught up, who may or may not be transformed.

What I cannot understand is that if all are destined to die, then it is a contradiction that believers will be caught up alive, without ever facing death. Though Enoch and Elijah were taken alive, some believe that they are the two witnesses mentioned in Revelation 11, who will be killed by the Beast, thus still fulfilling the words of God that all must die. 

Is there any possibility (apart from the scripture about Rapture), that some people will never face death?

Does God's word "Dust to dust" leave any exception?

How do theologians explain this contradiction?
The following statement is directed specifically to Adam as only Adam technically came from dust!  We of course as the offspring of Adam are affected relationally.  However, this statement would not be substantial enough to impose death or the &quot;returning to dust&quot; upon every member of humanity.

By the sweat of your brow you will eat your food until you return to the ground, since from it you were taken; for dust you are and to dust you will return. (Genesis 3:19, NIV)

Paul uses the word koimaō (Strongs G2837) in both of the following verses.  Notice that the word is translated as “dead” in 7:39 and could not be appropriately translated as sleep.  Conversely 15:51 could easily have used the word died in place of sleep.

1 Corinthians 7:39 The wife is bound by the law as long as her husband liveth; but if her husband be dead, G2837 she is at liberty to be married to whom she will; only in the Lord.
1 Corinthians 15:51 Behold, I shew you a mystery; We shall not all sleep, G2837 but we shall all be changed,

Hebrews 9 is not about how many times a man dies, but about the one time sacrifice of Christ.  “Appointed once to die” is in accordance that Christ does not have to continue to be crucified for our sins in the way the Temple sacrifices had to continue to die.  Essentially it is saying that it is natural for a man to die once.

Hebrews 9:24-28 For Christ is not entered into the holy places made with hands, which are the figures of the true; but into heaven itself, now to appear in the presence of God for us: Nor yet that he should offer himself often, as the high priest entereth into the holy place every year with blood of others; For then must he often have suffered since the foundation of the world: but now once in the end of the world hath he appeared to put away sin by the sacrifice of himself. And as it is appointed unto men once to die, but after this the judgment: So Christ was once offered to bear the sins of many; and unto them that look for him shall he appear the second time without sin unto salvation.

In Hebrews 9:27 the Greek word &quot;hapax&quot; is translated as &quot;once&quot;, it could just as well be translated as &quot;conclude&quot;, certainly not a strong enough statement to make a claim of contradiction.
It is also generally accepted form scripture that Elijah (Kings II Chapter 2, Verse 11) and Enoch (Genesis 5:22-24) did not die.
Therefore, While it is generally true that each man dies once, it does not mean that all men must die once.  Therefor, these verses are not in conflict with &quot;Rapture&quot; theology.