Holy Apostle James(...) When the Savior began to teach the nation about the Kingdom of God, St James believed in Christ and became His apostle. He was chosen as the first Bishop of Jerusalem.
So an Apostle can also be a bishop, there is no dilema.
Yes, but the earliest lists given by Irenaeus and Eusebius do not include Peter. Therefore, it simply is not part of the tradition to claim that he was.
b Peter the Apostle, by nation a Galilean, first high priest of the Christians, after he had been the first to found a church at Antioch, proceeded to Rome, where as bishop of the same city he remains, preaching the gospel for 25 years.
English translation of Jerome's Chronicon - book 2 of Eusebius with Jerome's additions.
That would make the first person Jerome, not Eusebius (who as Cavaradossi has pointed out starts counting Roman bishops from St. Linus
No it would not make Jerome, because this book is from Eusebius, only the notes are from st Jerome.
I simply didn't find online the book of Eusebius without Jerome's additions.
But i believe that the fact that St Peter was a bishop is a pious Tradition of the Church:
"but that greatest of disciples among disciples, and of teachers among teachers, who presided and ruled over the Roman Church, and held the chief place in the priesthood as he did in the faith. "
On the Incarnation (John Cassian) > Book III http://www.newadvent.org/fathers/35093.htm
To understand what st John Cassian means by priesthood:
"And finally, now that the mystery of this Divine priesthood has descended to human agency, it runs not by the line of birth, nor is that which flesh and blood created, chosen, but without regard to the privilege of paternity and succession by inheritance, those men are received by the Church as its rulers whom the Holy Ghost prepares: so that in the people of God's adoption, the whole body of which is priestly and royal, it is not the prerogative of earthly origin which obtains the unction , but the condescension of Divine grace which creates the bishop."
Sermons of St. Leo the Great > Sermon 3 http://www.newadvent.org/fathers/360303.htm
This is confirmed later in the same sermon by st leo:
'And hence the presence of my venerable brothers and fellow priests, so much desired and valued by me, will be the more sacred and precious, if they will transfer the chief honour of this service in which they have deigned to take part to him whom they know to be not only the patron of this see, but also the primate of all bishops. "
Here he calls st Peter the primate-bishop and confrims that Rome was and still is Peter's epicsopal see.
"African synod in opposition to Stephen,41444144 Bishop of Rome from May 12, a.d. 254, to Aug. 2, a.d. 257. See note on ch. 25. who was the blessed Peter’s twenty-second successor in the see of Rome."
The Dialogue Against the Luciferians. St Jerome; http://www.ccel.org/ccel/schaff/npnf206.vi.iv.html
"But the highest prince of the Apostles fought with us: for we had on our side his imitator and the successor in his see"
Holy Fathers of the Ecumenical Council of Constantinople III
"In his see" means an episcopal see, meaning st Peter's episcopal see.
"You cannot then deny that you do know that upon Peter first in the City of Rome was bestowed the Episcopal Chair [Cathedra], on which sat Peter"
"Well then, on the one Chair, which is the first of the Endowments, Peter was the first to sit."
Those 2 quotes are from st Optatus of Milevis, Against the Donatists; http://www.tertullian.org/fathers/optatus_02_book2.htm#C2
So yes, i think both catholics and eastern orthodox can agree that the Tradition accepts and confirms that st Peter was not only an apostle, but also a Bishop.
And i think that the Holy Scriptures also agree with this Tradition. Peter 5:1, Peter refers to himself as a “fellow elder” or presbyter.
"Presbyter and episkopos appear to be used interchangeably at other points in the New Testament and in I Clement. This usage makes it difficult to accept the thesis that Peter would not have been considered a bishop or episkopos while resident in Rome." www.catholic-convert.com/documents/PeterInRome.doc