Was The Apostle Peter The First Pope? (A Biblical Response)

Roman Catholics claim that the Apostle Peter was the first pope.  They base this assertion on several passages in the Bible, but primarily on Matthew 16:16-19.  

(Matthew 16:16-19)"Simon Peter replied, You are the Christ, the Son of the living God.  And Jesus answered him, Blessed are you, Simon Bar Jonah! For flesh and blood has not revealed this to you, but my Father who is in heaven.  And I tell you, you are Peter, and on this rock I will build my church, and the gates of hell shall not prevail against it.  I will give you the keys of the kingdom of heaven, and whatever you bind on earth shall be bound in heaven, and whatever you loose on earth shall be loosed in heaven." (ESV) 

When Peter said Jesus is the Christ, the Son of the living God, he was speaking on behalf of the other apostles as well.  The rock on which the church is built is Peter's confession that Jesus is the Christ the Son of the living God, not Peter himself.  This is the confession of all true believers.  Jesus says, "on this rock" referring to the confession of Peter.  He does not say on you Peter I will build my church.  

Also Catholics believe the giving of the keys of heaven was meant for Peter alone as pope and not for the other apostles.  But in Matthew 18:18 this authority is given to all the Apostles, not just Peter.  

Peter never refers to himself as the pope, in fact in 1 Peter 5:1 he refers to himself as a fellow elder.  Peter also refused to be worshiped and said I too am a man (Acts 10:25-26).  The apostle Paul never refers to Peter as pope and he actually rebuked Peter in Galatians 2:11-14.  This would be strange conduct from Paul if Peter was the infallible Pope.  At the Jerusalem Council in Acts 15 Peter functions as one of the apostles and not as pope, while James seems to direct the meeting.  

The office of pope is found nowhere in the Bible.  The papacy is not Biblical and therefore has no authority.