Q1: The article refers to "internal arguments" and "external arguments". What is the difference?
A1: In the general context of historiography, internal arguments are those which only analyze a document on its own, e.g. look at the tone or phraseology of a passage and compare it with the entire work, etc. The external arguments may rely on comparison with the works of other authors about historical dates, etc. and go beyond the document being analyzed.
Q2: Does the New Testament (say Acts 12:2) refer to "the death of James" as Josephus does?
A2: No, it does not. That is why the article states "Christian tradition" when discussing that issue. And the footnote explains that the unrelated New Testament reference is to James, son of Zebedee, who is killed by King Herod in Acts 12:2 with a sword.
Q3: How does the difference between the account of Josephus about the death of James and the Christian tradition indicate authenticity?
A3: That rationale does not just apply to this passage or Josephus but is used by historians in a more general context. The reasoning is that a Christian scribe would have been unlikely to differ from the Christian tradition and would have likely interpolated items to agree with it. Similar reasoning is used elsewhere about the "negative tone" in passages about Christians as indications of authenticity, in that Christian scribes were unlikely to be derisive of their own traditions.
Q4: Why does the article state that the "overwhelming majority of scholars" hold the James passage to be authentic? Did we do a survey ourselves?
A4: The formal Wikipedia guidelines require us not to do our own survey. The Wikipedia guideline WP:RS/AC specifically states: "The statement that all or most scientists or scholars hold a certain view requires reliable sourcing that directly says that all or most scientists or scholars hold that view." Given that the guideline then states: "statement in Wikipedia that academic consensus exists on a topic must be sourced rather than being based on the opinion or assessment of editors." we should not rely on our own surveys but quote a scholar who states what the "academic consensus" may be. That is what the article does.
Q5: The article states that most scholars hold that the Testimonium Flavianum had an authentic kernel that referred to Jesus, but was enhanced later. Are these just the Christian scholars?
A5: No. That "most scholars" agree that the Testimonium had an authentic core is supported by a variety sources, e.g. the highly respected Jewish scholar Louis Feldman, as well as the leader of the the 20th-century myth theorists G. A. Wells who acknowledges that after the discoveries of Shlomo Pines in the 1970s most scholars support that view. (The Jesus Legend, 1996, by G. A. Wells, ISBN 0812693345, page 48)