Prophet Muhammad

Additionally, it does not make sense to directly compare the process of compilation of the Bible and the Qur'an because the two scriptures are quite different structurally and in terms of the historical circumstances surrounding them.

The Qur'an is much newer than the Bible, much shorter than the Bible, emerged in one region and in a short span of time. There has never been any question of what portions of the Qur'an to consider canonical. It was also all in a single language and dialect. Stylistically, the Qur'an is internally consistent indicating it is the work of a single author and does not have interpolations (whether or not one considers that author to be God, it still has a single "voice"). While, at the time of the Prophet, Mecca and Medina did not have sophisticated means of producing books, it was not long before advances in those areas, such as the use of paper, came to the Islamic world; and, in fact, it was through the Islamic world that some of these technologies were transferred to Europe. 

In contrast, the compilation of the Bible is much more complicated. The scriptures are much older, span a broader length of time, and there is a lot more complexity with respect to how it came to be a single text. (There are numerous works on this subject available.) It becomes even more complicated if one considers both the New and the Old Testaments. It is not as if someone 2000+ years ago held out a book (or, rather, a scroll) called "The Bible" that is exactly the same as you find today; there was a process. 

So, from a purely historical perspective, there are very different circumstances surrounding the compilation of what today we know as the Bible and the Qur'an, and the nature of the two books is also quite different. Therefore, it is not a one-to-one comparison and it is best to look at the compilation of each scripture separately and consider the validity or authenticity of each one separately. 

The first part of this question is really a question about the authenticity of the Qur'anic text: is it all as the Prophet Muhammad stated, or could anything have been added to it over the centuries?

Since, from the time of the Prophet, the Qur'an was heavily memorized and was transmitted by multiple transmitters, and was compiled early in Islamic history, it is unlikely that anything was added to it, as it would not have been accepted by the people. There was also a lot of strictness in copying the Qur'an. (This is in contrast to other historical manuscripts, which may easily have had interpolations) It is virtually impossible to find anyone who seriously argues that anything was added to the Qur'an after the time of the Prophet Muhammad.

For more information on the compilation of the Qur'an, you could consult Qur'anic Sciences by Abbas and Masuma Jaffer.

(From an Islamic theological perspective, the entire Qur'an is the word of God not specifically the Prophet, so technically this verse is seen as the word of God not the word of the Prophet; for the Prophet's style of speech, one could consult the hadith.)

As for Christianity itself... Certainly Christianity IS a world religion that is acknowledged as an existing religion in the Qur'an. There have been many interpretations of Christianity and ways that it is lived, and there were a number of sects in and around the Arabian Peninsula.

The Qur'an praises some aspects of Christianity and acknowledges it a religion of God, while at the same time it criticizes some beliefs or gives a different viewpoint. This is one of those differences and clearly it is a major theological difference, but it doesn't mean that Christianity as a whole has no truth or merit to it.

While the Qur'an teaches that all prophets before the time of Islam, including Jesus, taught the same message, it does not say that every religion that emerged was absolutely correct or identical. Even just considering Judaism, Christianity, and Islam, there are obviously major differences between them! Rather, the position of the Qur'an is that some people deviated from the teachings of their prophets or introduced incorrect beliefs or practices, and this is why there is variance in religions.