How many Greek manuscripts in total do we have from the second and third centuries? How does this compare to all other non-biblical manuscripts?
I found what seems like reliable information at wikipedia that gave the following numbers for Greek New Testament manuscripts. https://en.wikipedia.org/wiki/Biblical_manuscript
2nd century 2
late second/early third century 6
third century 30
late third century/early fourth 10
fourth century 29
Those fourth century manuscripts include entire New Testaments, such as the codex Sinaiticus, the codex Vaticanus and the codex Alexandrinus. In the second and third centuries most of the manuscripts are on papyrus, which does not preserve well. For the fourth century and onward most of the preserved manuscripts are on vellum/parchment, which is much more easily preserved.
As a comparison, this website tells us that there are 476 known manuscripts of all types from the second century. Let us give a VERY rough guess and say that somewhere between four to eight percent of all preserved manuscripts from the second and third centuries are from the New Testament, which is astounding considering the relatively small role that the church played in the Roman empire in those centuries. The number of preserved manuscripts goes up exponentially in the fifth century and beyond, of course, because it is at that time that Christianity became the dominant religion in the empire.
Of course, all historians of both the Near East during Old Testament times and the Greek/Roman world of the first century use the Bible as source material. It is the most reliable source of historical information we have from these periods. We have the histories of the Greeks Theucidides and Herodotus as well as Romans such as Suetonius and Tacitus, as well as Jews such as Josephus, and these are very helpful, but these authors are often biased toward a particular political point of view. Bible writers have their prejudices as well, of course, but they tend to deal honestly with the information.