You might find this discussion useful to answer the question about where and how the text originated.

There aren’t good theories as to who wrote the Quran if Muhammad wasn’t the author (or transmitter, in the Islamic worldview) — some people have their favorite candidates, but none are proven.

Since you’re coming from a Biblical Studies background, you might check out the work of Gabriel Said Reynolds — he’s a prof at Notre Dame, and he’s done a lot of interesting comparative work on the Bible and Qur’an (it’s academic, not polemical - he’s more interested in the message and meaning of the religious texts).

We have relatively early witnesses of the Qur'an like the Birmingham Quran.

The traditional story, that still somewhat persists in historical circles AFAIK is that it was "revealed" to Mohammed bit by bit in verses that dealt with contemporary concerns and then finally compiled (not in chronological order) in the caliphate of Uthman, who burned all other textual variants. If you're looking for a likely writer or compiler beyond that (like say...the exilic or post-exilic community for the Torah, with Mohammedan authorship being a later attribution like Mosaic authorship) I don't know that there is one.

However, recently there's been a more revisionist tack taken by Stephen Shoemaker in Death of a Prophet questions this (in addition to the whole traditional story and the sources). In Shoemaker's view he doesn't challenge the authorship by Mohammed but the final compilation is more likely closer to the time of Abd Al Malik and then projected backwards into Uthman's reign. Though even he claims that:

The Islamic tradition professes that the Qurʾān was collected into its present form just twenty or so years after the death of Muhammad, during the second half of the caliph ʿUthmān’s reign (644–56), a tenet that has been adopted almost wholesale by most modern scholarship on the Qurʾān and Islamic origins

Shoemaker may or may not be right, but some of the challenges him and revisionists raise are important and do touch on the comparisons between the two histories.

Islamic studies seems to be in an interesting place when it comes to the historicity of stuff within the life of the Prophet. The Qur'an isn't really a biography of Mohammed; it has a very allusive style . So our early witnesses don't necessarily close the issue. A lot of historical data is within the hadith and sira traditions, which show up a century or more (in some cases two and a half) later and have their own issues in terms of dating and accuracy, to say the least . Depending on where you fall along the more-less revionist scale all of this alleged history can be up for grabs.

Since you're trained in Biblical studies I will try to point out Shoemaker's(or Crone's, since he seems to draw it from her)comparison to New Testament studies here.

So, compared to the New Testament, we're in a weird space. The Qur'an has better early witnesses than the Gospels but the surrounding traditions that provide the context are compiled or written much later than ones we have from the early Christian community. It's questionable whether the methods used for analyzing the writings of the latter can apply to the former.

The differing approaches to Christian and Islamic origins can perhaps be brought into greater relief by extending an earlier observation by Patricia Crone in the “Historiographical Introduction” to her Slaves on Horses. Here Crone completes her insightful analysis of the sources available for studying earliest Islam through a comparison of the sīra traditions with similar material from a variety of different cultures, including the earliest biography of the Buddha, the Icelandic sagas, and the Christian gospels. Regarding the latter in particular, she asks in a footnote that we “consider the prospect of reconstructing the origins of Christianity on the basis of the writings of Clement or Justin Martyr in a recension by Origen.”6 With this brief remark, Crone succinctly captures the stark difference in the quality of the sources available for the study of earliest Christianity and earliest Islam. In essence, Crone asks that we imagine that the earliest extant gospel was written not by “Mark” sometime around 70, roughly four decades after the crucifixion of Jesus, but rather by Clement of Alexandria (ca. 150–ca. 211), and moreover that this earliest biography survives only in an edition revised by Origen of Alexandria (ca. 185–ca. 254). For the sake of comparison, one should perhaps presume that in addition we actually have a copy of the lost “Q” gospel, a collection of sayings with no narrative context or passion narrative that was compiled approximately twenty to thirty years after the death of Jesus. The Q gospel reveals no information regarding the death of Jesus, nor really any knowledge of his activities or their setting, making it roughly analogous to the Qurʾān.7 For this information, we would depend entirely on this hypothetical “Gospel according to Clement” as now revised in the “Gospel according to Origen.”

He goes on and on. The contrast he paints between the two fields of historical inquiry and their sources is stark, to say the least.