A quick point here regarding Ploutos' third point on "doctrinal continuity and doctrinal development" -- be aware of John Henry Cardinal Newman's theory of development of doctrine. Many of us in the Reformed world who know Calvin's Institutes and/or the Reformation-era debates with Roman Catholics are using a type of argumentation that won't work with a theologically and historically knowledgeable Roman Catholic.
Both Protestants and Catholics suffered significantly in the 1500s with lack of reliable and factually accurate historical information about both ancient and early medieval church history. That's not anybody's fault -- documents we have today either weren't translated into the common vernacular academic language of the 1500s (Latin, rather than Greek), or they were hidden in some ancient monastery library and didn't get found until centuries later. In other cases, historical frauds or honest mistakes were widely believed to be true by both sides of the debate.
That led to Roman Catholics of the 1500s and even the 1600s and 1700s defending the argument that Catholic doctrine and practice had not changed, or at least had not significantly changed, from the ancient world to their time. Much of the historical section of Calvin's Institutes is devoted to proving that the Roman Catholic Church had significantly changed both its doctrine and its practices, and that some of those changes weren't ancient but of recent vintage (i.e., the early medieval period, only a few centuries earlier than Calvin's day).
I did my senior thesis on Cardinal Newman and I'm well aware that his theory is complex. The short version is that by the 1800s it was no longer possible to argue that many of the major Roman Catholic doctrines had been unambiguously taught in the ancient church or even the Nicene era, i.e., the 300s. Newman didn't argue that full Tridentine orthodoxy was taught in the 100s and 200s. He knew better. What he argued was that the Protestant principle of sola scriptura would logically lead to the teaching of Arius as the Nicene Council, not to trinitarian orthodoxy, and that while papal authority was certainly weak in the ancient world and nothing like later developments and claims, ancient church fathers, including St. Augustine, deferred to Rome in matters of major controversy.
Newman's argument wasn't that doctrine had been constant since the 100s or 200s, but rather that doctrine develops over time. He argued -- and he had a point -- that the Methodism of John and Charles Wesley developed over time from being a pietist movement of reform within the Church of England to become a separate denomination with numerous splinter groups, but that if we study the teaching of the Wesley brothers, we can see quite easily which of the Wesleys' successors were faithfully developing the core doctrines of their founders and which were deviating from them.
He argued that in each major controversy of the church, whether the Arian controversy or the Pelagian controversy or the various medieval controversies, we can look to earlier church history to see which of the two parties was logically and consistently developing the doctrine of their predecessors.
Again, Newman had a point. Those of us who are Van Tillian presuppositionalists will understand the argument that people will be consistent with their core beliefs and presuppositions, and that doctrines have consequences, and if we accept certain principles, they may take us places we did not anticipate and where we do not want to go.
Calvin and Zwingli were both willing to say that some of their doctrines broke with (in their day) twelve to fourteen centuries of church history. They argued that if we can't find something in the Bible, we shouldn't find it in the church. In that, they were quite a bit more radical than Luther.
I believe the Reformed were right. And if I agreed with Cardinal Newman's view that Arius was accurately applying Sola Scriptura, I'd be a Catholic today.
Newman was wrong about Arius. But if we grant his point, many consequences will follow, including that we must rely on church history and church councils to teach us what orthodox doctrine is.
This argument isn't what Calvin or Luther confronted. It's different, and needs to be responded to differently.
With both EOs and RCs, the best you can do with the historical record is to argue that it is not monolithic and that there is historical support for Protestant and Reformed positions. But you'll have to face three obstacles behind that: 1) it is often the case that historical support may be stronger on some issues for non-Protestant views; 2) RCs and EOs not only claim that history is on their side, but they also claim the authoritative right to interpret history; 3) they will obfuscate on the difference between doctrinal continuity and doctrinal development.
Ultimately, while some of the arguments can be answered, others can't and shouldn't be answered because they're framed in ways that concede certain foundational beliefs inimicable to Protestantism. So you're just going to have to be prepared to flat out reject some of their claims and to not get roped into arguing with them on their terms. This is why ultimately you not only need to be grounded in the teachings of the Bible but also in the doctrine of Biblical authority and supremacy as our only infallible rule of faith.
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