the veil, was in fact an Arabic cultural tradition, originally.

It's my understanding that the concept of married women veiling their faces in public was a cultural feature among elites in both Byzantium and Persia in the near-east before the rise of Islam that existed in pre-Islamic Arabia but did not become a near-universal practice of Islamicate culture until after the conquests. However it's not a topic that I've studied in any great detail.

I think your overall question however is probably too broad to answer in any kind of succinct fashion. Many things changed. Many things stayed the same. As in all places, culture shifted over time. There's a very good reason that the primary journal of historical sociology is called "Continuity and Change." In the Middle East before and after Islam there was continuity..and there was change.

I don't mean this answer to be patronizing but perhaps you could narrow down the topics, time periods and specific locations you're interested in. For instance, even if region-wide religious practice trended over time towards Islam, the change that that entailed in previously Christian territories was quite different than in previously Zoroastrian territories. And so on and so forth.