Practices are not doctrine. A doctrine is a tenet of faith, not a liturgical rite, a canonical court, or a rule of worship (e.g. fasting before reception of Holy Communion).
The Inquisition certainly was not a doctrine. It was an an investigation into heretical practices and imposed punishments upon perpetrators of heresy. Recent scholarly research (and I will find it for you, if you wish, since I cannot recall the author or title at this moment) discovered that the Inquisition's reputation as the agent of thousands of murders is patently and provably untrue. That some injustices were perpetrated, yes, that is probable. But the number of tortures and executions was probably less than 100 persons. In any case, the Inquisition is not a doctrine.
That there is only one, apostolic, catholic church has been a church doctrine since the earliest days of the church, as was the belief that outside the church there is no salvation. You can find this doctrine in Counciliar documents from the fourth century, for example. The Nicene Creed is from that era, and the idea of only one church is quite clearly present in it.
Purgatory is found in the canon of scripture that was defined by the church in the late fourth century. Protestant scriptural revisionists removed those books from their version of the scriptural canon in the 1500s because they inconveniently supported the doctrine Protestants did not want to believe.
Oral Tradition was the repository of sacred scripture until it was collected into the Bible as defined by the Council of Chalcedon (a Catholic council, since at that time no other church existed!). Since the Holy Spirit protects the oral, teaching Tradition of the Church, from which flowed the Holy Scriptures, Tradition (with a capital "T") enjoys the same inerrancy as Scripture. There are stringent qualifications for doctrines that reside in Tradition that make them infallible.
The Immaculate Conception of Mary is an ancient belief, found in early Church fathers' writings. Mary was always honored as the Ark of the New Testament, who carried the Son of God in her womb. She was the source of Jesus' human genes, his human component. To give the Son of God the perfect Ark, God preserved Mary from the stain of original sin, something the Almighty could easily do. Mary gained special grace from God to preserve her sinlessness throughout her life. Again, this doctrine was not an innovation: it was formalized as a dogma of the Faith because innovators of belief wanted to change what the Church had always believed and taught.
Which brings us to infallibility. I cited in my first response to you the fathers of the Council of Chalcedon (late 400s), who adopted the theology of Leo I with the acclamation "Peter has spoken through Leo!" This acclamation of the bishops at the council is but one example of bishops recognizing the special teaching authority and protections bestowed upon Peter and his successors by Jesus. Jesus conferred upon Peter the Keys of the Kingdom: "What you bind on earth will be bound in heaven; what you loose on earth will be loosed in heaven", and in so doing Jesus ensured an ultimate authority in His church. Again, this was not a novelty; it was an ancient belief, called into question by innovators, and the Church had to clarify and codify what it had always taught.
The Assumption of Mary is also not a new belief. Mary's purity and God-graced sinlessness preserved her from bodily corruption, and she was assumed into heaven by God, as an honor for the mother of God. In other words, all these things did not just "pop up" a thousand or more years after Christ's death. They were all present in some form or another since the beginning.
I hope this helps.
Regards