Nope. This is flagrantly not true. Find me a single historical source that teaches that Rome had singular authority over the catholic/universal church in the early centuries. Far more often you'll find people deferring to the bishops of Alexandria and Antioch, or to the ecumenical councils, than to any singular authority of Rome. Even then, it still doesn't go far enough to prove your main argument.
I reply: The Churches throughout the world were all CATHOLIC! All answered to Rome!
Being catholic is not the same thing as being Roman. There is a difference.
As for answering to Rome, this is not true. Rome was an authoritative centre, yes. But did the other churches answer to Rome? Not any more than Rome answered to the other centres. Again, you ignored the problems of the synods of Carthage and Hippo, councils that produced important canons that were only later sent to Rome for their approval.
FACT: Until 1054 A.D. there was JUST the Catholic Church headed by popes and BISHOPS!
Question does your church have Bishops with AUTHORITY to teach!!?
Again, these were not exclusively Roman. catholic =/= Roman. And yes, I have bishops. I don't see how, in and of itself, that proves anything.
"Owing to the sudden and repeated calamities and misfortunes which have befallen us, we must acknowledge that we have been somewhat tardy in turning our attention to the matters in dispute among you, beloved; and especially that abominable and unholy sedition, alien and foreign to the elect of God, which a few rash and self-willed persons have inflamed to such madness that your venerable and illustrious name, worthy to be loved by all men, has been greatly defamed. . . . Accept our counsel and you will have nothing to regret. . . . If anyone disobey the things which have been said by him [God] through us [i.e., that you must reinstate your leaders], let them know that they will involve themselves in transgression and in no small danger. . . . You will afford us joy and gladness if being obedient to the things which we have written through the Holy Spirit, you will root out the wicked passion of jealousy" (Letter to the Corinthians 1, 58–59, 63 [A.D. 80]).
This doesn't prove anything. Church leaders who were not Roman would often write to others outside their own local context. The Letter to the Romans is one example - it seems fairly clear from the letter that Paul is writing to a community of believers in existence before his contact with them, and yet he exercises authority over them as an apostle (cf Romans 16, amongst plenty of other places in the letter). Thus, your quote from Clement of Rome does not in itself prove anything
Hermas
"Therefore shall you [Hermas] write two little books and send one to Clement [Bishop of Rome] and one to Grapte. Clement shall then send it to the cities abroad, because that is his duty" (The Shepherd 2:4:3 [A.D. 80]).
As above. The fact that he sent letters abroad means nothing, as plenty of early church figures sent letters abroad. If, as is likely, Herma was a Roman Christian (if we go by Paul in Romans 16), then it makes perfect sense without having to even go near conjecturing a first century papacy.
"Ignatius . . . to the church also which holds the presidency, in the location of the country of the Romans, worthy of God, worthy of honor, worthy of blessing, worthy of praise, worthy of success, worthy of sanctification, and, because you hold the presidency in love, named after Christ and named after the Father" (Letter to the Romans 1:1 [A.D. 110]).
"You [the church at Rome] have envied no one, but others you have taught. I desire only that what you have enjoined in your instructions may remain in force" (ibid., 3:1).
This is a rather interesting translation you have picked up on. Your translation seems to imply that the church that hold's the presidency (seemingly universal) is in the country of the Romans. However, the Greek (ἥτις καὶ προκάθηται ἐν τόπῳ χωρίου ῾Ρωμαίων) does not necessarily intimate that. Instead, it more likely reads (and this is how Lightfoot renders it, for example) that the presidency (such as it is) is exercised by this church in the region of the Romans. It does not necessarily indicate that this primacy was held elsewhere, and even if it does, it is not clear from Ignatius whether this is meant to be a doctrinal authority, or simply being foremost in charity/love (Paul describes the Corinthians in similar ways, when he talks about them being 'first' in various things). Again, this doesn't reach the benchmark.
"For from the beginning it has been your custom to do good to all the brethren in various ways and to send contributions to all the churches in every city. . . . This custom your blessed Bishop Soter has not only preserved, but is augmenting, by furnishing an abundance of supplies to the saints and by urging with consoling words, as a loving father his children, the brethren who are journeying" (Letter to Pope Soter in Eusebius, Church History 4:23:9 [A.D. 170]).
As did the Corinthians, and the Macedonians. I don't see what you think this proves.
"Today we have observed the Lord’s holy day, in which we have read your letter [Pope Soter]. Whenever we do read it [in church], we shall be able to profit thereby, as also we do when we read the earlier letter written to us by Clement" (ibid., 4:23:11).
As did Paul, as did various church leaders from across Christendom. You don't think people were not widely reading the works of Papias, or Origen, for instance? I could use this same argument to argue for the primacy of any number of churches in the early centuries.
[The Martyrs of Lyons
"And when a dissension arose about these said people [the Montanists], the brethren in Gaul once more . . . [sent letters] to the brethren in Asia and Phrygia and, moreover to Eleutherius, who was then [A.D. 175] bishop of the Romans, negotiating for the peace of the churches" (Eusebius, Church History 5:3:4 [A.D. 312]) [/quote]
You notice how letters were sent here to more places than just Rome? We'll also ignore the fact that Rome would, obviously, have a greater episcopal oversight of Gaul, France being then under the jurisdiction of the Western Roman Empire.
"But since it would be too long to enumerate in such a volume as this the succession of all the churches, we shall confound all those who, in whatever manner, whether through self-satisfaction or vainglory, or through blindness and wicked opinion, assemble other than where it is proper, by pointing out here the successions of the bishops of the greatest and most ancient church known to all, founded and organized at Rome by the two most glorious apostles, Peter and Paul, that church which has the tradition and the faith which comes down to us after having been announced to men by the apostles. With that church, because of its superior origin, all the churches must agree, that is, all the faithful in the whole world, and it is in her that the faithful everywhere have maintained the apostolic tradition" (Against Heresies 3:3:2 [A.D. 189]).
Do you notice what Irenaeus is doing here, though? He is using as an examplar Rome, because it would take too long to discuss the succession of all the churches. That is what he says. So already, he is making the point that the other churches had DIFFERENT LINES OF SUCCESSION. He certainly holds the second century church of Rome up as a good exemplar, but the whole reason he is discussing it actually argues AGAINST considering it as the one authority of the universal church.
Also, the last line of the translation you have provided seems dubious. I couldn't find a Greek text, but I found a Latin one
here, which at that last line actually says something much closer to "it is necessary that every church should agree with this church (i.e Rome) on account of its eminence (connected to the ministries of Paul and Peter), inasmuch as the faithful everywhere have preserved the apostolic traditions." Nothing about the apostolic tradition being preserved specifically and only through Rome (which would make the next two pars Irenaeus writes complete nonsense) This seems very different to what you posted, and it seems that basically Irenaeus is arguing that the church in Rome is a good church, and every church should be agreed in unity with it in so far as all are agreed on the traditions of the apostles.
It's very interesting to me that Irenaeus then goes on, after paragraph 3, to then discuss in paragraph 4 the person of Polycarp of Smyrna, who was also a student of the apostles, and provided the apostolic succession in the Asiatic churches. In other words, Irenaeus' basic argument is "The heretics are wrong, because in all places, there has been an accurate succession from the apostles. There are two key examples - Rome (the West) and Asia (the East). These have both independently ensured apostolic succession, and are true witnesses to the gospel." Worth noting he also mentions Ephesus briefly as an aside.
To argue that Ignatius is somehow arguing specifically for the primacy of Rome here is to ignore most of what he writes. It seems to require a kind of logical gymnastics to make him say that when he talks about the pre-eminance of Rome, he means it is in charge of the other churches. It is an exemplar, yes, as is Smyrna. The pre-eminence argument, though, has to be read in the context of why Ignatius is writing this in the first place - he is arguing in favour of the orthodox churches on the basis of their apostolic tradition. This being the case, the point is less whether Rome is pre-eminent in favour of other churches, but rather that it is pre-eminent in favour of the heretical teachings (cf Against Heresies 3:4:3, where the above discussion of the churches in Rome and Smyrna is contrasted with the teachings of Valentinus and Marcion - these teachings originated with their namesakes, whereas the teachings of Rome and Smyrna originated with the apostles).
I'm still not convinced. I think what I would need to see would be multiple church fathers putting Roman pre-eminance, specifically in terms of Petrine succession. Otherwise, any position Rome could have, like Alexandria or Constantinople, could be a function of its social status and financial resources, rather than for any theological reason.