As far as i know, people in ancient Rome weren't buried within city limits, but always outside the city, near roads.
I know Peter was crucified there, but why would they make an exception for someone who was publically humiliated by crucifixion by giving him the 'honor' of burying them within city limits afterwards?
Or is it much simpler and did they move his grave (bones) much later to where the Vatican now lies?
I'm not an expert on archaeology, but I can tell you a bit about the excavations on that site. Here are some amazing pictures from LIFE magazine. Most are from 1950, and a few illustrate the problems they had with water invading the preceedings. The excavations began due to Pius XI's desire to be buried as closely as possible to St. Peter. They continued after Pius XI's death in 1939 into the pontificate of Pius XII, and they took nearly a decade to complete. Their conclusion was that St. Peter's Cathedral was built on the site of the tomb of St. Peter.
To answer some of your direct questions, the site apparently wasn't inside the original city limits and was a cemetery. Now, the Vatican's conclusions about the site was that St. Peter was not completely undisturbed, having been removed from his original grave on the site and placed in a special-built location. There were inscriptions and mosaics that identified the location as important to early Christians.
I haven't seen any substantial refutations of the idea that this was St. Peter's tomb, but I haven't looked for them either so I could easily be out of date on the subject. My expertise is purely on the motivations for the excavations and the fact that they continued straight through WWII, and that they were important to both Pius XI and his successor Pius XII. The rest of my knowledge comes from a few trips to Rome nearly two decades ago, and were all from the Vatican itself, so there is an admitted bias there. You can visit the tomb yourself if you're ever in Rome.
Keep in mind that a church was built on the site from the fourth century and the current St. Peter's Cathedral was begun in the 1500's.
(Edited to add the following)
Let me be clear--I do not know if St. Peter is buried below the Vatican. The excavations were important to Pius XI, which subsequently made them important to Pius XII. The evidence is far from definitive.
I'm not well-versed in the history of the early Christians, but I can tell you that it was perfectly fine to bury someone on the Vatican Hill. The taboo against burial within the city only applied to the city in its ritual sense, that is the boundary marked out by the pomerium, the ritual limits of the city. The pomerium was quite small, only enclosing a couple of the old hills, and activities which were illegal inside the city could be done usually in parts of the city just outside the boundary. This applied to burial as well, and the two largest necropoleis at Rome are found on the Campus Martius and the Vatican, both outside the pomerium but very much part of the city