1) I just reread 1 Corinthians 9 in it’s entirety, and it still looks to me like Paul is talking primarily about the ministry of the Gospel and the rewards associated with that as the basis for his personal discipline against his earthy appetites. The context is definitely not individual salvation. If you're convinced otherwise, make a detailed argument, but please don't ask me to accept a general assertion that totally ignores multiple obvious references to Paul's ministration of the Gospel and the rewards associated with that ministry.
2) I don't do genealogical arguments, and Petrine succession is precisely that, and necessarily so. I will instead direct you to a good discussion which I am relatively certain you will consider old hat. Be that as it may. I don’t do infinite loops either. I have a day job, after all. Perhaps when I retire … Anyway, here’s the site:
How Reliable Is Roman Catholic History?
3) I will address your point on authority. All Christians do have a single point of unified authority, the risen Jesus. He is not hypothetical, but more perfectly real than any institutional counterpart could ever be. Furthermore, he is not incompetent to administer and preserve his own church, and the idea that some human intermediary must be there to help him coalesce the exercise of authority in his church runs contrary to the risen Jesus pictured in the Apocalypse, as one who was busily rewarding some churches for their faithfulness and threatening others with dissolution, all without so much as a whisper of Peter's voice or office, but as something he was quite able to do on his own. Believers have no need for any master but Jesus (as he said), because His sheep will always hear his voice (as he said). Peace, SR
Jesus founded His Church on St. Peter and I am not one to argue with God on this matter! I can prove it historically and through Scriptures. There was never a dispute of this fact in the first centuries of the Church...
“It is to Peter himself that He says; “You are Peter, and upon this rock I will build My Church (Matt 16:18).” —St. Ambrose of Milan, On Twelve Psalms 40:30+. J1261, 387 A.D.
“I follow no leader but Christ and join in communion with none but your blessedness, that is the chair of Peter. I know that this is the rock on which the CHURCH has been built. Whoever eats the Lamb outside this house is profane. Anyone who is not in the ark of Noah will perish when the flood prevails...He that is joined to the chair of Peter is accepted by me.” -—St. Jerome, Letter to Pope Damasus 15:2, J1346,1346a, 374 A.D.
“In the same reign of Claudius, the all good and gracious providence which watches over all things guided Peter, the great and mighty one of the Apostles, who, because of his virtue, was the spokesman for all the others to Rome” -—Eusebius, History of the Church 2:14:6. J651dd, 300 A.D.
You can follow Calvin’s teaching, that is your choice; but I will stick to St. Peter and the Church which has withstood 2000 years of abuse from both within and without. This longevity is only possible through God’s divine hand.
Peace.