Oh man, am I inspired here. I've picked up some books on early Karaism and looked through some academic articles (thank god for university digital library collections) on interactions between Islamic scholars and Jewish thinkers during the early Caliphate. Since the ATL equivalent of Mu'tazilism is both much older and is more-or-less the reigning interpretation of Islamic kalam, there's going to be a really interesting transferal of ideas between the two groups. I talked to one of my professors about this (very obliquely, I just asked a series of what-ifs instead of trying to explain this hobby
) and she said that in a mixed Rashidun Army like TTL would have, the military itself would probably become the most progressive institution in the Caliphate - being both the home of barracks cultural admixture and the Muslims who travel the farthest from home before settling down. Though there will be strong caliphs, meh caliphs and caliphs who almost wreck the institutions of the Caliphate, I think the long period of Soldier-Caliphs that follow the Companion-Caliphs will be a period of building on early reforms, the preservation of the Principate-esque quasi-republican nature of the Caliphate, and the forging of a Caliphal identity (similar to Romanitas) out of the many peoples in the empire.
I'll be clear that the TTL Caliphate isn't a pluralist paradise or anything - although minority faiths literally having a seat at the table, serving in the army which is the pride of the Caliphate, and having the ear of the Caliph is nice compared to the old Sassanids or the Eastern Romans, Muslims are given preferential treatment with the stratification between Muslims on the top, Jews and Christians in the middle, and Zoroastrians on the bottom still remaining.
Despite all of this, the idea that there might actually be a peaceful Jerusalem with real and thriving communities of Jews, Christians, Muslims, and Samaritans in the TTL Caliphate is something that really makes me happy.
And don't worry, Caliph Umar will still build the Dome of the Rock. Commemoration of the Is'ra wal-Mi'raj doesn't seem like something that would be butterflied away
(although TTL's Dome of the Rock might feature very intricate images of Muhammad meeting the various prophets of the Old Testament and Jesus in the heavens.)