Reread the text. The keys are the power to bind and loose. In Matthew 18, he gives this same power to the church (not the centralized organization, but the body of all believers). Verse 1 speaks of the "disciples" not the "apostles" per se.
Then, IN CONTEXT: Jesus says later in the same chapter,(Matt 18)
15Moreover if thy brother shall trespass against thee, go and tell him his fault between thee and him alone: if he shall hear thee, thou hast gained thy brother.
16But if he will not hear thee, then take with thee one or two more, that in the mouth of two or three witnesses every word may be established.
17And if he shall neglect to hear them, tell it unto the church: but if he neglect to hear the church, let him be unto thee as an heathen man and a publican.
18Verily I say unto you, Whatsoever ye shall bind on earth shall be bound in heaven: and whatsoever ye shall loose on earth shall be loosed in heaven.
19Again I say unto you, That if two of you shall agree on earth as touching any thing that they shall ask, it shall be done for them of my Father which is in heaven.
20For where two or three are gathered together in my name, there am I in the midst of them.
Peter then asks a question regarding forgiveness and Jesus starts telling him about forgiveness and how it is in the Kingdom of Heaven.
Similarly, in the Keys of the Kingdom passage (Matt 16) Jesus says: 19And I will give unto thee the keys of the kingdom of heaven: and whatsoever thou shalt bind on earth shall be bound in heaven: and whatsoever thou shalt loose on earth shall be loosed in heaven.
Again, when the RISEN Lord showed himself to the DISCIPLES that had gathered He said:
John 20:23Whose soever sins ye remit, they are remitted unto them; and whose soever sins ye retain, they are retained.
This was BEFORE there was an organized church anywhere. After this, the church spread like wildfire. Paul travelled most extensively, but Peter's words seem addressed not to the Romans but to those in Asia Minor. It can not be proven that Peter was ever in Rome and his epistles, for someone who is supposedly the 1st Pope are somewhat narrow in focus. They mainly deal with encouragement to the persecuted church and wasn't a doctrinal letter per se like Paul's letters were.
Again, if he were the first Pope, why not mention it? Even hint it? No special exclusive power was given to Peter to do anything. And Peter is NEVER EVER referred to as the head of the church on earth. He was one of the Lord's great apostles. And JESUS Himself is head over the Church.
Why would you assume that? The Keys are described as the keys to the kindgom if heaven. The power to bind and loose is listed separately in Chapter 16. In Chapter 18 the keys are not mentioned, yet the authority to bind and loose is given. While the two are related, there is no warrant from scripture to equate the two.
not the centralized organization, but the body of all believers
I always wondered where in the scripture do people find that strange ecclesiology. The Catholic view is that the boundary of the Church is baptism. It is therefore visible Church; there is not such thing as Church invisible. The apostolic hierarchy of the Church is unseparable from the entire Church. The text you cite in Matthew 18 is clear that the Church, and not the fellowship of believers is the final authority in any dispute.
It can not be proven that Peter was ever in Rome and his epistles, for someone who is supposedly the 1st Pope are somewhat narrow in focus.
It sems at this point beyond doubt that he was (see St. Peter and Rome). Peter's letter establish the apostolic authority with which he and the rest of the apostles speak. This is the fundamental task of the papacy, to preserve a single deposit of faith free from fracture.
if he were the first Pope, why not mention it?
The word "Pope" is of later extraction. His exceptional authority is mentioned in the Keys passage and in several others. It is true however that the power and the role of the papacy grew in post-patristic period as the heresies were combatted. Much of the papal power is derived from the Pope's role as patriarch of the Latin Church, since for most people today the Latin Church is THE Catholic Church. But the Catholic Church as a whole has a far more conciliar structure than the Latin Church.