Question 1: Is it not true that we don't even know in what century Jesus lived? How come we only have a lot of references in the New Testament and no where else from that general time?
Answer 1: You will have to work pretty hard to find scholars who argue the thesis that Jesus never lived. Even most "liberals" dismiss these views as baseless. It has been refuted time and time again. Why? Because there are first century references to Jesus, several of which critical scholars date to within months to a couple of years after Jesus' death. I'm speaking here chiefly of the early creeds in the New Testament, like 1 Corinthians 15:3ff. Besides all of the New Testament writings, we have a few extra-biblical writings that date from the mid-first century to about 110 AD. Altogether, there are even about a dozen and a half non-Christian sources that mention Jesus within the first 150 years after his death. For all these sources plus a critique of views like those who question or deny Jesus' historical existence, see my book The Historical Jesus (College Press, 1996).
Question 2: Is it true that Josephus' statements about Jesus are in fact not his and were added later in history by those seeking to prove that Jesus was a historical figure?
Answer 2: The vast majority of scholars who address this issue think that although Josephus' longer statement about Jesus in Antiquities 18:3 has been altered a bit, the bulk of it was written by Josephus. This view means that Josephus supplies some very important material about Jesus. An even larger percentage of scholars accepts Josephus' second statement concerning Jesus being the brother of James (Antiquities 20:9). Further, we have to make sense of ancient non-Christian historians like Thallus, Tacitus, Suetonius, and Lucian, who reported all sorts of facts about Jesus. In The Historical Jesus, pages 243-250, I provide a long list of well over 100 items that are reported about Jesus, many by non- Christians. So, to argue that Jesus never existed totally ignores a large body of historical data. That's why, of over a thousands recent publications on the subject of the historical Jesus, I am aware of less than five who doubt or question his existence.
Question 3: Why do you suppose Josephus does not discuss Jesus in even more detail? Assuming from his two passages that he was in fact aware of Jesus and the corresponding movement, isn't it a bit odd that he includes no other discussion on Christianity? There is plenty about John the Baptist, Pilate, Caiaphas, etc., but very little about Jesus.
Answer 3: I don't suppose anyone knows exactly why Josephus doesn't say more about Jesus than he does, or why, more generally, any writer doesn't say more about someone, especially in ancient times. One possibility could be that Josephus catered to his Roman patrons, and of course, they crucified Jesus. For instance, neither Tacitus, nor Suetonius, nor Pliny the Younger speak well of Christianity. All of them, by the way, along with Josephus, clearly place Jesus in the traditional time slot. But given this general reluctance not to laud Jesus (Pliny states that early Christians sang hymns to Jesus as to a god and even says that he killed Christians who failed to worship the gods), it's not terribly surprising that Josephus doesn't say more.