I read in Robert Betts' Christians in the Arab East that Edessa in pre-Islamic era was a major cultural center for Christianity, especially for Syriac Christianity (the Jacobite one).
This isn't explored further in his book, so I gather a few information from the internet and found that its fringe position (between Byzantium and the Arab desert) made it not completely Hellenized and was a hub for trading. But that's all about it.
How exactly was the process that it became the major cultural center in the 6th century? And if Edessa was a major Christian center, how was the religious life in the city? How was it compared to Antioch or Alexandria?
How exactly was the process that it became the major cultural center in the 6th century?
Indeed: Edessa became one of the major cultural for Syriac christianity because of its fringe position and because of its old tradition. Imperial control and thus demand for orthodoxy wasn't as tight as it was in Antioch or Palestine. The local narrative claimed that the king of Osroene, the ancient client kingdom which separated the Roman empire from Parthia, wrote a letter to Jesus and that one of the seventy-two disciples went to Edessa to baptize the king. While this story which is passed on by Eusebius is to be dismissed it shows that the local church in Edessa had a high standing in the 4th century. The most important author in ancient Syriac relocated together with the school of Nisibis after the Persian conquest of Nisibis in 363.
And if Edessa was a major Christian center, how was the religious life in the city?
Edessa was together with Nisibis the major centre for Christian of the Syriac tradition. From writings we know that there were many monasteries and churches. I can't tell you any more details, it is not my area of expertise. I'd like to add, though, that the Christian history of Edessa is in a difficult spot. It is situated in Turkey and the local Christians were exterminated or evicted after the first world war. The political elite in Turkey is currently blocking all efforts to investigate these events. I don't know of any archeological projects on ancient Edessa in the last hundred years. Other than the Armenians who now have a country to operate from, so to speak, the "Assyrians" as the Christians call themselves now don't have the resources to undertake efforts to preserve their history.
Suggested further reading (haven't read it myself):
J. B. Segal, Edessa: 'The Blessed City', Oxford 1970. (Bit pricey, unfortunately)