The short answer is yes.
A longer answer is that the study of the Qur’an by non-Muslims is something of a complicated subject in Islam. While the Qur’an recognizes that it will be of interest to non-Muslims, its self-identity as religious truth makes challenging it akin to denying that truth. Readers, then, are segregated between those who accept its truth and those who do not, and are therefore antagonistic. There is not really an idea of someone reading the Qur’an for purely secular enlightenment. Early Islam tended to see the motivations of interested but non-converting parties as antagonistic and there is evidence of things like early Christians being forbidden to teach their children the Qur’an on the grounds that they would twist the truth.
Over the centuries, this distrustful view was pretty heavily validated by Christian apologetics that used the Qur’anic text to attack the supposed truth of Islam. It was not until the later middle ages when understanding of the Qur’an began to be seen by Christian scholars as useful for its own sake, without worrying about apologetics or converting. You began to see works that were not particularly polemical and driven by a desire to understand, even if the authors were almost invariably hostile to Islam itself.
When the historical-critical studies of the Bible came into vogue in the 18th century, it took a little while for it to be applied to the Qur’an because scholars still had this idea that the Qur’an was derivative of the Bible. There was, however, some very interesting philological scholarship done in Germany around this time that marked a big shift in European Islamic studies. Abraham Geiger (19th C) wrote a very interesting work that sought to trace the origins of the Qur’an within Judaism and Christianity, and did so without raising the “truth” or “false prophet” issues that other works did. It placed Muhammad, as best it could, within the context of his time. Authors like Gustav Weil and Theodor Nöldeke continued in this vein. They did a good job in some ways and applied a lot of tools you mention, but they were a bit too text-focused for me.
This kind of approach was basically what Western scholarship focused on up until 1970. What was interesting, though, is even when scholars like Richard Bell approached the Qur’an from new directions and were critical of Islamic claims, they still maintained acceptance of basic tenants of Muslim traditions of the Qur’an and its compilation, treating these historical accounts, at times, more seriously than Muslims themselves. So you get this idea that the origins of the Qur’an are uncontroversial and settled and it was quickly compiled and was caused by, if not revelation, then at least a religious experience by Muhammad, etc.
Dissatisfaction with this state of affairs led to a flowering of new scholarship that has really pushed a lot of the bounds of Qur’anic inquiry and mainstream thought. One pivotal and controversial author is John Wansbrough, who directly challenged Islamic historical traditions and proposed a greatly extended formation period for the Qur’an. Wansbrough, along with later author John Burton, labeled Muslim sources of early Islam forms of literary activism rather than history. He took his lead from biblical studies and identified independent traditions that seem to imply a much larger and longer pool of origin for the Qur’an than tradition would say. The specific Qur’an we have today, Wansbrough said, was the result of 9th century Islamic scholars seeking to justify their legal rulings, rather than an Abu Bakr collection or an Uthman one.
While Wansbrough is not exactly taken as gospel (heh) today, his work’s biggest impact was in opening up the study of Qur’an to new methods, assumptions, and a bit of creativity. Angelika Neuwirth contributed but went on quite a different path, looking at the literary structure of the Qur’an from the base unit of the Sura rather than a complete work as Wansbrough did. Her research understands the Qur’an as a collection of liturgical units developed for worship purposes and she tracks their evolutions in stages as chronicling the rise of the proto-Islamic sect as an independent group.
A couple more examples of innovative, if at times going near polemical/controversial, research in the Biblical studies mode: Gunther Luling, who was preoccupied with the idea that Christian and Jewish dogmas had abandoned primitive beliefs, took to the study of Islam and stripped the Qur’an pretty much down to its bare bones in order to conclude that the Qur’an has a pre-Islamic Christian text as a primitive bedrock layer. So there are formerly Christian passages that have been given new meanings by the tradition, and new Islamic passages put on top of them. It was also influenced by pre-Islamic Arab Paganism. Luling holds that the Qur’an was revised several times in the course of reaching its final form. My recollection is Luling is considered to have been pretty reaching at times in making his conclusions, though there was definite Christian influence on Early Islam and the composition of the Qur’an.
Cristoph Luxenbourg wrote a very interesting study on how Syro-Aramaic influenced the Qur’an’s Arabic. He believed the language of the Qur’an was a mixture of Aramaic and Arabic and that the knowledge of the peculiar Meccan dialect spoken by Muhammad was lost by the time punctuation and exegesis of the text began, which is why there was a lot of confusion and misreading of Muslim scholars. As you can imagine, this interpretation has not gone unchallenged. He gives a period of about 1.5 centuries for the Qur’an to fully form, and is notable for downplaying the importance of the Qur’an in the earliest days of Islam.
These guys are all a bit old, but I hope they show the trajectory that Islamic studies have gone in terms of not being afraid to push the envelope. I don’t know newer stuff on this as well but I’m sure I can dig some out if you see it necessary. Patricia Crone is fun. Karen Armstrong is nice too. Reza Aslan has some interesting insights.
Recently (last 30 years) there has been a growing movement among Qur’anic (and actually, all sorts of religious scholars) scholars to stop focusing on text so much and examine instead how Muslims themselves have approached the Qur’an, both in practice and in exegetical literature. Focusing too much on the circumstances and historical origins of text removes us from the actual realities that these texts inspire. In this sense, then, the study of the Qur’an is moving along pretty well with the study of other religious traditions and texts. There still is a (not always called for) tendency towards polemics, and on the other side, a (again not always called for) tendency in the Muslim community to view scholarship produced by non-Muslims as risky and untrustworthy at best. But there’s a ton of good scholarship being produced in the meantime.
I do not necessarily endorse the views of the above authors, though I will be glad to expand on them if anyone cares. They are just representative (I hope) of movements in how Western scholarship has approached the Qur’an and how the answer to your question is basically yes. There is a lot we don’t know about early Islamic history, a lot to debate, and a lot of research being done.
If I have made any mistakes in the above, I look forward to correction. To refresh my memory in writing this I consulted the Cambridge Companion to the Qur’an, Chapters 3 and 11, by Harald Motzki and Andrew Rippin, respectively. To make it simpler on myself I kept it to mostly info from there, which explains why I chose the above authors to mention.