Has the Qur'an been written many times in ways different from what we see today? Are the letters present in it the same as the original?

Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

The noble Qur'an was transcripted during the life of the Prophet (Sallallahu Alaihi wa Sallam).  The Prophet (Sallallahu Alaihi wa Sallam) used to have scribes who wrote the revelation for him.  Every time verses of the Qur'an were revealed, he would say to them: "Put this in such or such place, in such Surah."

When the Prophet (Sallallahu Alaihi wa Sallam) died, the noble Qur'an had been copied on leather skins, cloth materials, wood boards,….etc.  But the Qur'an not had been gathered in one place, though many companions of the Prophet (Sallallahu Alaihi wa Sallam) knew it by heart. 

Now, after the Prophet's (Sallallahu Alaihi wa Sallam) death, many of the companions who knew the Qur'an died in Al Yamama battle.  Thereupon, the companions feared that part of the Qur'an would be lost because of the death of those who knew the Qur'an by heart, so they decided to collect the whole Qur'an and write it by all available means at that time.  This was during the era of Caliph Abu Bakr (Radiya Allahu Anhu).  Such collection was the first after the death of the Prophet (Sallallahu Alaihi wa Sallam).  The Caliphs (Muslim rulers) kept the things on which the Qur'an was written with them.

Finally, the existing Qur'an Book (Musahaf) was put together in one book during the era of Caliph Othman Ibn Affan (Radiya Allahu Anhu).  Its transcription and its letters never changed since then.  This is why, we notice that some people write on the Qur'an Book the expression "According to Othman's transcription".  This means that it is according to the spelling in which the Qur'an was written during the era of Caliph Othman (Radiya Allahu Anhu).

These are the two times when the Qur'an was collected.  Each time it was with the same letters, words and spellings without any addition or reduction or deformation.  The Qur'an cannot be changed.  Allah Says (interpretation of meaning): {Verily, We; It is We Who have sent down the Dhikr (i.e. the Qur'ân) and surely, We will guard it (from corruption).}[15:9]. Allah also Says (interpretation of meaning): {Falsehood cannot come to it from before it or behind it (it is) sent down by the All-Wise, Worthy of all praise (Allâh).}[41:42].  

Allah knows best.