Was Peter the first pope? If so, does this prove Catholicism?
Catholicism can neither be proved nor disproved. You cannot “prove” a religious group. Catholicism has beliefs and practices which are biblical and Catholicism has beliefs and practices that are unbiblical. I am sure this is true of all Christian groups.
One of the unbiblical practices of Roman Catholicism is that of having a “pope.” The only “pope” the Christian church has is Jesus Christ! The Bible does not anticipate a single human leader overseeing all of Christianity. Quite the contrary.
About Peter, he definitely was not a pope in the Roman Catholic sense. Historians will debate who was the first true pope. Some say Gregory in the sixth century. Others say Leo in the fifth century. The papacy developed over time. Roman bishops as early as the second century were claiming higher levels of authority over churches in the West. By the third century, they were beginning to claim some sort of preeminence among the bishops, but with very little acceptance of the claim. By the fourth century, Roman bishops were starting to exert significant authority in the West and were, arguably, the most powerful of all the bishops. It was not until the fifth century, however, that Roman bishops such as Leo were claiming anything close to the kind of universal authority over all of Christianity that we see today. Even then, they were not called papa (pope).
Either way, Peter was not a Roman pope. It is not confirmed historically that Peter ever led the church in Rome. Even if he had (we are not sure) he would have been a bishop, not a pope. He almost certainly was in Rome in the 60s AD and was killed there. To claim that he was the first pope is to deny the facts of history. By the fourth century, bishops in Rome began to publish lists of successive bishops in Rome which included Peter as the first bishop. This was historical revisionism, and even then the list was not yet a list of popes but of head bishops.
On your second question, whether Peter was a pope or not (he was not) and whether the Roman Catholic Church has popes now neither proves nor disproves Catholicism. There is no perfect Christian group. We should discern Christian groups according to how faithful they are to what Jesus taught and to what the Bible teaches. It is not uncommon in the West to do what I call Catholic-bashing. Despite the rather obvious unbiblical practices of the Roman church, this is not helpful. We should use the Golden Rule and begin by pointing the finger at ourselves, not at others (Isaiah 58:9)