Were clerics allowed to interpret the Quran? How open was the state to academic pursuit in Greek philosophers like Plato and Aristotle? Was there internal policing of the Academics in the society? Were people with foreign ideas allowed to visit?
The most influential medieval philosophers of the Islamic tradition worked in the time of the Abbasid caliphate - Al-Kindi, Al-Farabi and Avicenna, especially the latter, sent earthquakes through Islamic thought and his interpretations of Aristotle hugely influenced future thought; whether they were attacking his position or defending/expanding it, future Islamic philosophers could not ignore Avicenna's work.
Avicenna's most controversial ideas were a result of his work to try to reconcile Aristotelian philosophy with the Qu'ran. He was harshly criticised by later thinkers, specifically the titanic Al-Ghazali, who criticised him and other falasifa who leaned heavily on the works of Greek pagans like Aristotle and Plato. Avicenna wrote a long rebuttal to Al-Ghazali's work but ultimately proved less influential.
From just this brief snapshot, I think it's clear that the Abbasid Caliphate was a place where deep and fundamental disagreements could and did happen between hugely important thinkers. While the more conservative Al-Ghazali eventually triumphed, a strong tradition of interpreting the Qu'ran in a more Aristotelian vein had a long tradition in this period and others.