While in general agreement with your post, I believe that the Muslims do NOT consider that the Zoroastrians are 'dhimmi', people of the book, mostly because they were not. Zoroastrianism was an entirely separatge pagan religion, founded roughly in the same period as Buddhism, but in the Persian Empire. It was an ethical religion, and some of its beliefs passed into the Roman West, most especially the worship of the lesser Zoroastrian deity, Mithras. The symbol of Mithraism was a Sacred Bull, and that of Zoroastrianism was the fire.
While accepting Christians and Jews as followers of the same God, Islam gave the Zoroastrians the choice of converting or being executed. To this day islam has a horror of the 'fireworshippers'. As a result, not one Zoroastrian remains alive in the Persian homeland. A number of refugees fled to India, where they remain, and are known as 'Farsi', Hindu for a Persian. The Zoroastrians believe in a good creator god 'Ahura Mazda' and a nearly equally powerful evil god 'Ahriman'. They are known to their Hindu neighbors especially for their practices in desposing of the bodies of the dead. Instead of burying or cremating corpses, the Farsi leave the bodies exposed in walled courtyards, set aside for the purpose, where they can be eaten down to the bones by birds and other scavengers. This does not improve their reputation in the areas of India where they reside.
Not true. They are scarce and have suffered much persecution, but they still hold on. About 75,000 live around Yazd and in a few other places in Iran. This is not a whole lot less than the number in India (about 150,000), although in India they are a wealthy and influential group.
I think for the most part you are correct. They were not people of the book. As usual in Islamic lands however there were some exceptions, including with the people of the book notion such as occurred under the Almohads or other Islamic rulers. Hence, amongst the notables some Zoroastrians were allowed to continue and their related families. Indeed, as noted below, the movement of Zoroastrians to India occurred in the middle of the 10th century and hence they had been living a schizoid existence for over three hundred years under Islamic rule. This is what you also saw in India, Hindus not being people of the book also lived a schizoid existence at times conversion or die and other times as, effectively, dhimmis.
I thought they were monotheists.