I was under the impression that cataphract cavalry stopped being a mainstay in Iran after the fall of the Sassanids. Did Muslim Persians have such units in their regiments?
I believe you're right, they were probably used by the Umayyads in Perisa for a time but it seems there's little if any evidence to suggest widespread use by the time of CK3. The Byzantines would still make use of them at times, but it seems for the 867 start date they should be "softer" while being more akin to their late antiquity counterparts for the 1066 start if my reading of the Wikipedia article is correct.
I don't think so. Heavily armoured cavalry of various kinds never stopped being used in the region, including the 'fully enclosed' cataphracts. This applies to both the intermezzo and the seljuk periods, but also earlier.
A little bit of googling gives me this depiction of
khorasani heavy cavalry (heavily armored cavalier, no armor on the horse) this
other depiction of khorasani cavalry (fully enclosed horse armor, heavily armored rider),
this third straight up khorasani cataphract from an earlier post sassanid period, this depiction of
abassid persian cavalry (armored cavalier and horse, with cloth on top). All reminiscent of the
roman cavalry in various ways.
I think only centuries later by the Ottoman period that you see powers in the region opt for mobile light cavalry armies more and more (which is something that would eventually become prevalent in Europe as well), but you still had the
ottoman sipahis and the
mameluke turks making similar choices when outfitting their heavy cavalry. The
timurid cavalry is likewise reminiscent of cataphracts.
Of course you might have noticed there's a lot of variety here and there, but nothing was static. We are not just talking about the differences between auxiliary troops and core soldiers, but also centuries of polities and ways of waging war that vary over time. Rome didn't use the same cataphracts forever, but everyone tried to field the same sort of 'fully enclosed' heavy cavalry.
So really at the end of the day it's not that the arab invasion cleansed the persians of their ability to create cataphracts. Nor did the arabs turn the persians into a realm known for its iconic futuwaa spearmen. What happened is that the iranian empire was dismantled. The ability to field heavily armored cavalry diminished as local structures of political authority and revenue gathering disappeared. But the culture didn't change. Neither did the demands of the 'military meta' of the period, so to speak. Both the arabs and the rising powers of the period would go on fielding cavalry armies, as befit the region, the region's culture, and the region's military needs. With their ability to do so varying greatly over time.
The real changes in the time period isn't the death of the heavily armoured fully enclosed cavalry, but the stylistic changes introduced by the turks and the arabs, a growing demand in both Rome and the Caliphate for cavalry archer mercenaries (mostly turks and turkopoles), the rise of horse cavalry as a key military arm in general, and the Turks bringing their wacky curved swords with them.