3 Things to keep in mind.
Islam is to Iran like Christianity is to Norway. They adopted it, or were conquered, but they DIDNT help invent it, only "improved" it, as I like to say. Iranian architecture is NOT Muslim, but is still related closely with other Islamic architectural styles simply because these were influenced by Iranian architecture and vise versa.
IE Languages originated in Central Asia. It is thought that this is the ancestral home of all Indo-Iranians from Kurdistan to Bangladesh. The language family, (closely linked genetically as well, so this solves that question) migrated south and inhabited the regions previously inhabited by Mesopotamians, notably Elamites, or Dravidians(?) of Indus valley.
I am not sure of the genetic or linguistic heritage of the Mesopotamian peoples, nor am I certain of the Indus Valley civilization either, but I assume they are closely related to the Pre Indo-Iranian tribes and are related to the original peoples of India, the Dravidians. Indo-Iranians, many people from Kurdistan to Bangladesh are still the original people, for the most part, as most people still remain who they are after they are conquered. It is nearly impossible to remove a race or ethnicity from a region, and even then they will most likely migrate, and this did not happen in Iran, Afghanistan, Tajikistan. Rather, they were able to gain independence, while retaining Islam, and reconquer previously lost lands. The Samanids, and later the Safavids, were responsible for retaining some, if not most of this original "Perso-Iranian" culture.
Take a look at a Mosque from Iran and then one from Saudi Arabia and notice the stark differences (At least from the ones I've been able to find). When IEs entered the Iranian Plateau they adopted a lot of Mesopotamian cultural influences, i.e. mosaics and tiling, geometry, rounded turrets, and minarets. This was not found in Arabia, so we can notice the original Perso-Mesopotamian cultural influence.