I assume the quotes you talk about are the ones that started with Socrates' assertations. I will answer them one by one. In the meantime, however, I seem to miss your point on your answer to Antioch. Jerusalem's patriarchs succeeded from the line of St. James. Antioch's patriarchs were from St. Peter (no different from Rome).
Could you please elaborate?
Now on to the "quotes:"
Socrates wrote that Julius rebuked the Eusebians on the grounds that:
...it is unlawful to legislate for the churches without the consent of the bishop of Rome [HE II, 17 ]
Reading the text from here:http://www.ccel.org/fathers2/NPNF2-02/Npnf2-02-07.htm#P651_266599
I find that the quote however says something slightly different:
On the receipt of these contradictory communications, Julius first replied to the bishops who had written to him from Antioch, complaining of the acrimonious feeling they had evinced in their letter, and charging them with a violation of the canons, because they had not requested his attendance at the council, seeing that the ecclesiastical law required that the churches should pass no decisions contrary to the views of the bishop of Rome.
In the context of this quote, the Synod assembled at Antioch assumed itself to be of ecumenical status. But, as I have showed you, ecumenical status is not granted unless Rome gets in the picture. Rome wasn't even invited to begin with, let alone ask permission. We have seen examples from the patriarchs of Alexandria in the past, such as Pope Alexander and Pope Cyril, who took Rome's permission to lead ecumenical councils.
Therefore, Rome is mentioned due to something that required ecumenical attention. Rome is highest in honor because of the ecume, not because of St. Peter. There is no mention here of St. Peter as the cause of Rome's honor. It was the church canon that resulted in her honor.
Sozomen records that Julius criticized the Eusebians for failing to convoke him to the council at Antioch:
...because it is a law that actions taken without the consent of the bishop of Rome are invalid. [HE III, 10]
Taken into context from http://www.ccel.org/fathers2/NPNF2-02/Npnf2-02-21.htm#P3426_1451014
The bishops of Egypt,35 having sent a declaration in writing that these allegations were false, and Julius having been apprised that Athanasius was far from being in safety in Egypt, sent for him to his own city. He replied at the same time to the letter of the bishops who were convened at Antioch, for just then he happened to have received their epistle,36 and accused them of having clandestinely introduced innovations contrary to the dogmas of the Nicene council, and of having violated the laws of the Church, by neglecting to invite him to join their Synod; for he alleged that there is a sacerdotal canon which declares that whatever is enacted contrary to the judgment of the bishop of Rome is null.
This is the same exact situation. Sozomenus agrees with Socrates that Rome did not receive its honor that it was given by canon law, NOT by St. Peter. And this honor has nothing to do with individual churches, but the ecumenical churches together. The Eusebian Council assembled was not an ecumenical coucil, but by being contrary to Nicea, and not including the ecume, ESPECIALLY Rome, it was thus automatically null.
St. Theodore the Studite who called Rome:
... the chief throne in which Christ placed the keys of faith: against which the gates of hell, namely the mouths of heretics, have not prevailed up to now, nor shall they ever prevail, according to the promise of Him who does not lie [PG 99: 1281].
Forgive me. St. Theodore the Studite is not a saint in the Oriental Orthodox tradition. I would like to add however that Petrine primacy has been rejected in the fifth century ever since the Chalcedonian schism. There were fears that such primacy was lurking in the Church of Rome since then.
Perhaps, someone else more suitable than me can answer you on this quote.