Answering whether the "Qur'an has not been changed" is difficult, because not every Muslim will mean the same thing when they say it. I will try to look at different interpretations of this statement, but first a little bit of context.

Disclaimer that I will occasionally use ahadith, that are considered authentic by Islamic tradition but unreliable for the historical method, as sources for various accounts. But I think it's practical in this case, because Muslims believe in them and we are talking about what Muslims believe the questioned statement to mean, after all.

Muhammad supposedly died in 632, at which point there wouldn't have been any additions or redactions to the Qur'an according to tradition. He revealed the Qur'an orally and in Arabic. There are some accounts of Muhammad's companions keeping record in their own private codices. Modern scholarly consensus is that the majority of the Qur'an was not transmitted in written form during the prophet's life and that is was mostly memorized. There are some who believe that individually each verse was written down by at least someone, but never in its entirety in a single codex.

After Muhammad's death, the battle at Yamama took place in which 70 of Muhammad's companions who had memorized the Qur'an died. This led to Abu Bakr and Umar, Muhammad's successors, deciding to collect the Qur'an's scattered pieces, in order to confirm the memory of the remaining memorizers (and vice versa) and create a complete copy of the Qur'an^(a).

This task would take almost 20 years, and it included Uthman, Muhammad's third successor, canononizing the Qur'an. By the time this happened the Rashidun caliphate was vast and diverse, meaning that there were also different versions of the Qur'an. Uthman had his scribes make copies of the codex Umar relayed to his daughter and Muhammad's widow and spread those throughout the caliphate while ordering the destructions of deviating manuscripts.

Now, to the question, and I will try to keep this short because you could write books about even just individual interpretations of what the statement "the Qur'an is preserved" means.

Is the Qur'an eternal and can never be changed?

Depending on which Muslim you ask, you're going to get a different response. Generally the answers to this question can be split into two competing types of Islamic schools, reason and literalism. The school of reason would answer no, while the literalist would answer yes. But I can do you one better, there's actually some real world data. Dr. Marijn van Putten makes a identifies a few passages in the Qur'an to be interpolations, suspecting some of them to be autointerpolations, changes to the Qur'an made by the prophet himself. He spoke about it on a podcast recently, but the gist of it is that some of the verses break narrative and poetic flow, suggesting that they were abrogated or changed at some point, most probably during Muhammad's life. This opens up the idea that Muhammad changed his mind, or if you're a believer that either he made a mistake in transmission or that God changed his mind.

Is Uthman's Qur'an the same as the one that was transmitted by Muhammad?

Most of it yes, but almost certainly not all of it. There are dozens, if not hundreds, of traditional ("authentic") accounts of Muhammad's companions criticizing Uthman's canon as not being true to the original, most famously Ibn Mas'ud, who did not consider the first and the last two Sura's to be part of the Qur'an, and Ubayy Ibn Ka'b, who in some cases agreed with Ibn Mas'ud despite both his and Uthman's codex being mostly compiled by Zaid Ibn Thabit^(b).

Is the Qur'an we have today the same as the one Uthman canonized and distributed?

Yes, and no. The Uthmanic Qur'an was written with rasm only. The Arabic script is an abjad, not an alphabet, and requires diacritics to denote vowels. Rasm makes no use of diacritics, therefore vowels have to be inferred through context, which people did and it resulted in different readings of the Qur'an that still exist today. However, Dr. Hythem Sidky managed to prove that all of them can be traced back to a single shared ancestor, supposedly the Uthmanic/Meccan one.

Can the Qur'an still be understood the same way it was when it was originally conceived?

Sure, but that doesn't mean it is as Dr. Shady Nasser argues in his book "The Second Canonization of the Qurʾān (324/936)". He points to several points in time that he identifies as individual canonizations of the Qur'an, 4 in total including the Uthmanic one^(c), in which the meaning of the reading was restandardized.

Here's a lecture summarizing his research.

a: Muhammad Mustafa Al-Azami, The History of The Qur'anic Text: From Revelation to Compilation: A Comparative Study with the Old and New Testaments, p.83

b: Nöldeke, T. Geschichte des Qorans. Georg Olms Verlag, Hildesheim, Germany. 1981 (1909) (can't find the page anymore, I'm sorry)

c: Shady H. Nasser, The Second Canonization of the Qurʾān (324/936), p.258