Ok, so this covers the succession and the role of popes. What about the claim of the Pope being Peter's successor based on things like Matt. 16: 18-19? How are we to understand that verse and reply? I know of several scriptures and points used to refute this, but they are from Protestant sources so I don't know how Orthodox that understanding is.
Matt. 16:18-19, "And I also say to you that you are Peter, and upon this rock I will build My church; and the gates of Hades shall not overpower it. 19 "I will give you the keys of the kingdom of heaven; and whatever you shall bind on earth shall be bound in heaven, and whatever you shall loose on earth shall be loosed in heaven."
This is a very interesting topic, and there are different views on it.
I have seen two approaches in Orthodoxy. One is to show that the Bishops of Rome are not properly called St. Peters successors, and that even if they were, there is no reason that any headship St. Peter had, if he had any, should transfer to them.
The other approach is to admit that St. Peter did have a leadership role, but to show that every bishop is the successor of St. Peter, with his presbyters around him representing the Apostles. The Bishops are not successors of Apostles in the sense that they hold the same office as the Apostles. They are successors in that they carry on the mission of the Apostles, but there are differences in the offices. An Apostles is a roving preacher, who sets up bishops in every city. A bishop is the head of the community in that city, and may not minister outside his diocese, or move. The Apostles established the Churches, but they were not bishops of those cities, they were Apostles who kept moving on and preaching in the next city, leaving a bishop behind.
For this view, I would say see this paper: http://www.orthodoxanswers.org/media/documents/ecclesiology.pdf
Now, I think these two approaches are complementary, and each have something to say on the issue.
For the first approach, there is much that can be said. St. Peter founded Antioch, why isn't the Patriarch of Antioch infallible, just Rome? St. Peter did not even found the Church in Rome, St. Paul did, and then asked St. Peter to come help out with the Jews there, since St. Paul ministered to Gentiles, and St .Peter to Jewish believers.
Finally, for a third approach, why should we proof text from scripture, and infer from st. Peter being given the keys that the Pope or Rome must have certain faculties and authorities. This is quite a leap. It is necessary to consider history, and how the early Church operated rather than to do what the Protestants do, read the Bible, and try to reconstruct what the early Church looked like and how we should worship from the Bible, which is not an instruction manual on how to structure a Church, rather than looking at the writings of the Fathers and the early Church's history and practise.
I'm convinced that anyone who reads the first have of the first volume of the ANF + Eusebius' Church history, honestly and attentively, cannot accept Papal infallibility. The Bishop of Rome gradually trying to impose his will and extend his authority, with the rest of the Church ignoring him and rebuking him, can be seen throughout the early history. When a Pope or Rome excommunicated Asia minoir for not obeying him in his decree on the date for Pascha, Rome was rebuked by the bishop of the backwater Gaul. The power grabs are obvious, but no one ever bought the argument in the early Church.