First of all Allah the Almighty Himself has made a promise (HE even has sworn) to preserve it (15:9):
Indeed, it is We who sent down the Qur'an and indeed, We will be its guardian.
We only know about the human part of the preservation I'll quote what come in to my mind:
Tawatur this means Quran has always (from the time of sahaba on) been transmitted by many people to many people orally!
And you might not find a place on earth were at a time of the day Quran is not recited, as we are recommended by Quran itself to recite it in any situation any time of the day.
So the sum of this already should keep Quran always in memory. And by this should make it hard if not impossible to change the Quran.
Some conclusions form the story of the 1st Quran collection
The most reliable narration on the first collection of Quran is narrated on the authority of Zaid ibn Thabit himself in sahih al-Bukhari who even quoted some different wordings in other narrations (showing that this narration has some kind of tawatur) in the chapter on tafsir, in the chapter on virtues of the Quran and in the chapter on Judgments. At-Tirmidhi also quoted it in his Jami'.
This narration shows a few points:
Why was Zaid ibn Thabit chosen, as there have been other memorizers available and maybe even other scribes?
Note that Zaid was literate therefore he was one of the most known scribes of the Prophet, but there were many other sahaba who have been literate. As he was ordered to collect backups his own writings didn't count! We know from Quran that we always need at least two witnesses 2:282
... And bring to witness two witnesses from among your men. And if there are not two men [available], then a man and two women from those whom you accept as witnesses - so that if one of the women errs, then the other can remind her. ...
The major reasons are as stated in the narration:
...'You are a wise young man and we do not have any suspicion about you, and you used to write the Divine Inspiration for Allah's Messenger (ﷺ). So you should search for (the fragmentary scripts of) the Qur'an and collect it in one book). ...
- He was a young man, so he was able to do such a meaningful difficult duty, and also as a young man he wouldn't insist much on his own view (or recitation and records in this case), so in case of differences he might more easily accept a guidance or an advice. He therefore also was somewhat free to concentrate on this duty and if necessary travel to get copies of texts.
- He witnessed and was part of the recording of the divine text (at the life time of the Prophet), so he has seen what others might not have seen.
- He was considered as a wise (young) man, which made him predestined to accomplish this duty.
- He was an upright man (Abu Bakr said to him: "... and we do not have any suspicion about you...") he also was known for his devoutness, his trust and his good manners (as else our Prophet wouldn't chose him as a scribe). Also as such he wouldn't even think of changing the revelation as he clearly knows about verses like 2:42, 2:174 and 3:187.
- And he was known to be a memorizer who have memorized the Quran in the life time of the Prophet and he has been tested at the last time our Prophet "tested" the memorizers العرضة الأخيرة. this is quoted by many scholars among them Abdurrhaman as-Sulami, who quoted that Zaid recited the Quran twice in front of the Prophet. So Zaid exactly knew (or memorized) the final (version) of the Quranic text, as he was tested twice before the death of the Prophet , so he wouldn't confound it with revelations that have been abrogated! Some scholars even say he was aware of the 7 ahrouf, which is an extra qualification.
So Zaid with the guidance of abu Bakr and others has made his task somewhat difficult, but as it was a big task with a major impact on upcoming Muslim generation these precautions actually seem necessary:
So I started looking for the Qur'an and collecting it from (what was written on) palmed stalks, thin white stones and also from the men who knew it by heart, till I found the last Verse of Surat at-Tauba (Repentance) with Abi Khuzaima Al-Ansari, and I did not find it with anybody other than him. The Verse is: 'Verily there has come unto you an Apostle (Muhammad) from amongst yourselves. It grieves him that you should receive any injury or difficulty..(till the end of Surat-Baraa' (at-Tauba) (9.128-129) Then the complete manuscripts (copy) of the Qur'an remained with Abu Bakr till he died, then with 'Umar till the end of his life, and then with Hafsa, the daughter of 'Umar.
- He didn't relay on his recitation nor his records.
- He tried his best to get written records, to which he found enough witnesses who could testify these records to have been made at the time of the Prophet (preferably according to the dictation of the Prophet).
Note that this involved many sahaba in the act of collection which means an extra precaution for the task.
- Again even if he was a memorizer -and at the time reading or reciting with the seven ahruf was allowed- he was still searching for backup for his own memorization. And either found it in written form or memorized by sahaba who had witnesses of these being a Quranic text reveled and dictated or taught by our Prophet to them and approved as the final version of Quran (with no abrogated verses): This means at least three people came to Zaid and recited the same verses in the same "harf" in front of him.
So in each case we would have at least three if not four (counting Zaid as a witness) witnesses for that piece being part of the Quran, apart of it being later approved by other sahaba. There are even narration claiming that Zaid was not alone but he was accompanied by 'Omar (according a narration on the authorithy of 'Orwah ibn az-Zubair) and according to the Arabic wikipedia site al-Bukhari and at-Tirmidhi stated that Sa'id ibn al-'Aas, Abdurrahman ibn al-Harith and Abdullah ibn az-Zubair have been given the same task.
The narration shows that the last written verse Zaid has collected was the one from surat at-Tawbah. Here the translation plays a role, as the original text is really not allowing much interpretation whether there were other verses he didn't collect, or should have collected. The text says the last verse he was missing (as he knew it should exist) was the one from surat at-Tawbah, and as you may read it's neither the last verse in the order of the Quran, nor si surat at-Tawbah the last surah in that order, so it was more a like the last piece of puzzle which is needed to fill the "blank" somewhere in the middle. Narrations show that 'Omar was among the witnesses of this Verse being part of the final version of the Quran.
If there had been any objection to that, like it was for the 2nd Collection sahaba would have shown it, so one can assume that by the guidance of Allah and with his own efforts and persistence to accomplish the task he has been ordered to do the best way Zaid could collect all the pieces of the Quran and all the memorized parts too in a book which he gave once his task was finished abu Bakr. The reason for the 2nd Collection clearly show that "Quran students" had objections to wrong (pronunciation in the) recitations of their "teachers", so people of that time were strong enough in their faith to not stay calm and let things go if something may be wrong, you may find evidences of that in many ahadith and narrations.
In the 2nd collection the thing that created the most disapproval was restricting the Quran to one harf, this was strongly disapproved by Abdullah ibn Mas'ud (who said he recited the Quran 70 times in front of the Prophet, what ever this may mean, as it is hard to imagine all these 70 times to have taken place in the last year before the death of Muhammad) for a long time. Some scholars say his disapproval came because he was not asked to be a part of the comity for the compilation (he was among the first believers). So people of 'Iraq and especially al-Kufa rejected for a long time the recitation and the moshaf al-Imam with which 'Othman has sent Abdurrahman as-Sulami to them until 'Assim could find a middle way of a recitation between the one prescribed by the moshaf of 'Othman and the recitation of Abdullah ibn Masu'd (For details see my answer on Why is the Hafs reading of the Qur'an so prevalent?) and of course ibn Masu'd didn't die rejecting this restriction, but acknowledged it later.
The precautions taken by Zaid
- He didn't record any thing until the owner or transmitter could proof (by at least two witnesses) it was (part of the) Quran and it was recorded in presence of our Prophet.
Note that some sahaba have recorded the Quran beside some hadith.
- He didn't record any thing until he had a proof that it was part of the final recitation or "final version" of the Quran.
Note that only by this the following precaution can be confirmed:
- He didn't record anything until he had a proof that it was not abrogated by a newer verse or just deleted in the "final version".
Note that many of the records of sahaba included such revelations.
Some more explanation
Be aware that when our Prophet died there have been many people who memorized the (whole) Quran and and among them were sahaba like Zaid ibn Tahbit, Ubay ibn Ka'ab, Abdullah ibn Mas'ud, Abdullah ibn 'amr ibn al-'Aas, Salim mawla abi Hudaihfah, abu a-Drada' and Mu'ad ibn Jabal and many others whom apparently have died during the fights of those who left Islam after the death of our Prophet and we therefore don't know.
All of these sahaba were known to be memorizers of the Quran and you will find most of them (if not all of them) in the narrator chains of of the 7 or 10 (among Muslims) approved qir'aat! Some of these sahaba have later taught other sahaba who are also considered as hafidh or memorizers of Quran, but they at least were not memorizers at the time the Prophet died. And also as we are now many sahaba memorized parts of the Quran, some in presence of the Prophet or later from what he dictated those play a big role in the collection. As even if we assume there were only 10 or 20 hafidhs among the (much) more than 5000 sahaba who were still living at the time you might not find anybody who didn't memorize at least a part or a couple of verses of the Quran. And it's hard to believe that none of the later has not memorized it by (direct) oral transmission from our Prophet.
Can you imagine that among this amount of sahaba none would have objection if Zaid didn't accomplish his duty and left lets say two or three fragment pieces unwritten?
The duty of Zaid was not to write down the Quran as he memorized it, but to find a backup in written form of that what he memorized. And that's what he did, so when he was missing a verse he knew he did his best to find a text somewhere in the hands of one of the scribes or their family. And beside this he needed witnesses who could testify that he found this text with a person. In cases the memorizer or scribe who had recorded the words of Allah was already death and his family members might have split these pieces of text as a heritage between family members.
So Zaid kept searching until his "Quran-Puzzle" was completed and this puzzle was the first pattern for the copy which have been kept at the house of the mother of believers Hafsa after the death of abu Bakr and 'Omar. Note that some of the mothers of believers where also hafidh among them Hafsa and 'Aisha if i remember well.
Of course there are some claims about missing verses, but Quran has been transmitted by tawatur so single claims don't count especially as all sahaba have approved the copy of Zaid at the first collection and later the moshaf al-Imam of 'Othman. This means either those parts were in an early stage part of the Quran, but not of the final version which has been tested by Jibreel in the last Ramadan twice. Or they really were not a part of the Quran, as for example in Mushaf Ubay what we now know as "du'a al-Qunut" was compiled as a surah.
Also note that some sahaba have been known to have their own copies of the Quran among them as said Ubay ibn Ka'ab and Abdullah ibn Masu'd. As these moshafs have been destroyed or burnt beside any copy which has been made earlier like the (private) copies of some of the mothers of believers who have been based on the moshaf which has been kept at the house of Hafsa nobody can clearly say wether those "moshafs" have included the whole Quran, but we know for example from statements of the owners what they have recorded in it. For example ibn Masu'd said he didn't put neither al-fatiha inside nor the two mo'awidat (al-Falaq 113, an-Nass 114), because -as he said- if he would have recorded al-Fatiha he would ahve put it in fronte ach surah, and he understood the two mo'awidat as formulation to seek refuge by Allah from evil!
And finally you are wasting your time on a useless investigation as the Quran has not meant to be written in a book, but kept and memorized and transmitted orally from memorizer to memorizer in the way it was dictated or recited by our Prophet, as learning or memorizing the Quran from the book isn't enough and worthless, as only a person who has learnt and memorized (the whole Quran) from a hafidh (having a continuous chain of narration ending with our Prophet) can be considered a hafidh. So compiling it in a book was only a an extra precaution to preserve it in case that more memorizers die before transmitting their wisdom to other generations of memorizers. And having it recorded at the time of the Prophet was in first place to help them once a new revelation came to add it to the correct surah and for those who were memorizing it was a helping tool.
A person who memorizes the Quran from the book only is not called hafidh rather than calling him "moshafy", as he was never tested or he never recited the Quran in front of a hafidh who corrected his pronunciation or spelling or recitation in general and let him repeat it until he approved it.
Some Arabic sources that were helpful:
On the first collection of Quran.
How to explain that Muslims (apparently) have missed a verse of the Quran before the collection at the time of abu Bakr, based on the narration of ibn Shihab in the chapter of virtuous of the Quran in sahih al-Bukhari.
And finally here's a fatwa in English on the matter