First, welcome to the forum! I recommend that you first read "The Orthodox Church" by Metropolitan Kallistos (Ware). To answer your questions in a very simplistic way:
a. St Peter was not the first bishop of Rome. The Roman Catholic Church calls him that because of a singular interpretation of Matthew 16:18: "And I also say to you that you are Peter, and on this rock I will build My church, and the gates of Hades shall not prevail against it." in conjunction with the doctrine of Apostolic Succession, Rome interprets this to mean that the entire Church, everywhere, is subject to the Bishop of Rome, who is the only legitimate successor to St Peter, the first Bishop of Rome. This is a heresy from the Orthodox perspective.
b. However, from the earliest of times Rome was very respected amongst the ancient Patriarchates. First, because Rome was the first capital of the Roman Empire and second, because she was known as a being steadfastly orthodox. Rome was indeed the First among Equals and had a primacy of honor. That did not mean that her jurisdiction and formal power extended beyond the areas assigned to her. This situation was indeed the case during Saint Constantine's reign.