As far as I understand it, Muhammad's earliest instructions regarding wine, and the first such verse to be codified in the Quran, advised believers not to take part in prayer while intoxicated. It seems like quite a stretch to interpret th at as a complete ban on the consumption of alcohol.
Not only that, but many of the most renowned poets and intellectuals in the history of the Islamic world, such as Hafez, seem to have regularly indulged in wine. Even some Sufi mystics, typically known for asceticism, apparently drank wine.
Yet these days alcohol is hard to come by in most Middle Eastern countries, even those that are secular, due in part to a cultural lack of demand. How and why did this interpretation become so prevalent?
Regarding the first part of your question regarding the Qur'ān's take on alcohol, you're correct that the earliest instructions do seem to be strictly against praying while drunk (presumably a problem that needed to be addressed, but it is not completely prohibited). But, it doesn't end there. The Qur'ān also mentions that there is some benefit in wine (but also harm) in Q. 2.219, but then also that wine is an abomination and Satan's work in Q. 5.90.
The process of reconciling seemingly contradictory ideas in the Qur'ān is through the idea of abrogation -- that God's later revelations can nullify what came before. Some verses might be abrogated (mansukh), while others are abrogating (nasikh). So, the exegetes and scholars tried to figure out the occasions of revelation (asbab al-nuzul) as a means of creating a chronology for the Qur'ān's verses. Through that process, alcohol was determined to be 'an abomination' and 'Satan's work', though that was by no means a hard and fast rule that everyone in the Islamic world abided by. There is a long tradition of wine poetry, the recognition of monasteries being places for indulging in alcohol consumption (among other things), and lots of drinking all around.
So, that doesn't answer the whole question and I have to take off right now, but at some point later I might try to get back to this.