No, these are not fragments they are fully integrated into the oldest known full copies and are mentioned by Origen as far back as AD 230.
Origen complained that there were no references to Jesus the messiah in the texts. That should tell you something right there.
No, there are many scholars both Christian and non-Christian that do not think the entire passage was inserted just certain parts of it. But also there are some scholars that think the theological sections were part of the original and that Josephus was being sarcastic. There is nothing in textual criticism that points to it being inserted. The manuscript evidence is as unanimous and ample as it is for anything in Josephus. And this also includes his mention of John the Baptist which indirectly points to Jesus.
Most secular scholars believe the larger passage talking about Jesus is a complete insertion. Not only is the writing style not that of Josephus, the writing flows better if the passage is removed. The paragraph before talks about a tragedy that had hit the area, and flows perfectly into the next paragraph after the Jesus bit saying "about this time there was another sad calamity". The paragraph about Jesus sticks out like a sore thumb in the overall context of the chapter.
The only people who argue it's at least partially genuine are fundamentalist Christians. Virtually nobody with an unbiased perspective believes the paragraph is legitimate
No, the section about James is written long before he gets back to talking about Jesus the High Priest, thereby contradicting your statement below about Josephus style.
I'm not arguing that he wrote passages about James and Jesus, however they are James ben Damneus and Jesus ben Damneus. The James in question is not the brother of Jesus Christ, James was executed by Ananus, son of Ananus who was at the time high priest. Ananus was deposed because of the execution and the position was given to the brother of James, Jesus ben Damneus according to Jewish custom. That's what the whole passage is about. It doesn't fit the gospel story at all, Jesus Christ didn't ever become high priest of Israel, and if he had he certainly didn't get there because his brother was executed by the previous high priest.
Since the passage about Jesus occurs BEFORE the passage about James, this confirms the style you refer to. He explained who Jesus the "so-called Christ" was before he mentions his connection to James. There is no real evidence that the Jesus he is referring to the high priest.
He doesn't say "the so called Christ", the passage reads "who was called christ". That's something Josephus would never say. Josephus would have called him a charlatan or something along those lines. The passage about Jesus coming first is actually consistent with Josephus's style as he considered him the more important character. That makes sense as he was high priest of Israel.
That is why many scholars believe the original wording was similar to his passage about James, ie that he was the "so-called Christ" or he was "called the Christ". There is evidence that many ordinary Jews admired Jesus without accepting His message so Josephus may have admired Him for His good deeds and teaching but not agree with the Christians about who Jesus was.
Most unbiased secular scholars and many christian scholars believe the phrase "who was called christ" was likely a margin note which inadvertently got inserted into the text by a later christian scribe, and is not a deliberate forgery. However it was not original to the text.
It is strong evidence given it was composed within 5 years of His death and because of this most of the Romans and jews that had witnessed what had happened were still alive and would have refuted anything in the hymn that was not accurate. And there is no evidence that that happened.
There's no contemporary evidence at all that Jesus was executed, or even existed. If Jesus was indeed a real person, it's very likely he led a minor cult that basically nobody knew about and his death went largely unnoticed.
If he was the widely known popular miracle worker portrayed in the gospels, he'd certainly have shown up in the writings of contemporary historians. Philo of Alexandria springs to mind, he lived from 20BC to 50AD, meaning he was in his prime years right during the time Jesus was said to be running his ministry.
He lived in Egypt, however he was a Jew and often travelled to Jerusalem. In fact he had ties to the royal house of Judea. He's also regarded of one of the top historians in all of Roman history. His writings were actually quite popular with the early Christians as well who adopted some of his philosophical ideas into Christianity, such as the logos. Philo very literally could have been on scene during Jesus's ministry, seen the miracles, his trial, execution and resurrection. Certainly he would have mentioned something about him, even if he didn't buy that Jesus was divine or the messiah. There are plenty of other minor cult leaders mentioned in his writings, however Jesus doesn't even show up in that regard.
So what we're left with is a complete no comment from contemporary historians (Philo was the one who had the best access, however there were others who would have been in a position to write about Jesus as well), and at best we have two minor references from (at best) 60 years after the fact, that are very likely later insertions into the text.
That is the extent of non-biblical evidence for Jesus. It's absurd how weak it is if Jesus really was how the gospels portray him.