Coimbra1982 said
How relevant is the testimony or the writings of Yosef Ben Mattityahu (Josephus) to Christianity and specially to the historicity of the ministry and deeds of Jesus? Can it be considered to be authentic and reliable? Does he provide not only a reliable source but most of all a trustworthy one?   

The autobiography, Antiquities, and Wars provide historical context for the accounts in the New Testament.

Yes, these three works are reliable and have been used to inform us about the first century.

His manner of writing is NOT straightforward at all times. Second, there are some differences of the same characters and events when the stories are told in two or more of his works.

The biblical Jesus is not a unique biological human being, he is a composite character of historical fiction.