To start with Josephus, the Testimonium Flavianum is not a "known forgery", though it is *generally* thought that Eusebius added language that most (myself included) think doesn't sound like Josephus.
"Not a contemporary" of Jesus does not render these historians as worthless witnesses. They were not so far removed from Jesus' time that they could not easily find eye witnesses. I am not a contemporary of JFK, but I could find plenty of people who are, and write about what he did. And the fact that he died a few decades ago does not mean I can't write an accurate account. In fact, Luke states that he researched what happened so there could be a record for the followers.
Yes, people convert between religions all the time, but why? Usually because they weren't really convinced of what they said/thought they believed. Saul was a zealot, extremely knowledgeable and sold out for Judaism. Yet he suddenly converted to Christianity. Why? What causes a person to do a 180 like that? Likewise, most of the other apostles suffered horrible deaths rather than recant their claim that Jesus *did* die and rise from the grave. Why? I can understand if some follower hundreds of years down the line becomes convinced enough of what they are *told* that they die for it, but these men knew what was true, and they died for it.
J. Warner Wallace, a cold-case homicide detective, says that the differing details between the four gospels does not mean that they were made up, but quite the opposite: that they were accounts of different witnesses. If a lawyer questions four witnesses of a murder and they all tell exactly the same story, he knows that they colluded, and their story is highly suspect. On the other hand, if the overall story matches but details differ (were there two angels, or one? Was Peter alone or did John accompany him to the tomb?) the lawyer knows each person is telling his own account, as he/she remembers it. The lawyer can then piece together a cohesive account from their stories.
Christianity is an historical religion, and Jesus' historicity is its foundation. If that crumbles, the whole edifice falls with it. However, it has held for a long time against attack.