Short answer:

Its religious scripture. It can be an historical source, but like all sources, has to be examined critically to be useful to historians. There is no one judgment as to its "overall trustworthiness"; some parts are useful, some are less so.

Discussion:

Historians take a very different approach to a text than that taken by the faithful with their scripture. To an historian _any_ authentic historical source sheds light on something, whether what it says is "true" or not. Ancient forgeries -- of which there are many-- tell us something, even if they're not what they purport to be. People telling lies and exaggerating are useful too. For a believer, everything rides on their scriptures being "true" . . . not the case for an historian. Similarly, a believer doesn't care what contemporaneous texts _aren't_ in their Bible; the Dead Sea Scrolls don't change Lutheran or liturgy and doctrine one iota-- whereas they're enormously important for historians. Similarly, there are texts which are theologically important but which are of less historical significance-- the Book of Revelations would be an example. It doesn't purport to describe much of the world -- it's more supernatural and metaphysical.

Our sources inevitably include errors, exaggerations, lies, mis-translations, later edits and so on . . . an historian doesn't look for a "true" source, he looks for a useful source. Assembling a picture of the past doesn't mean deciding "aha, account X is true and account Y is a lie" . . . that's rarely the case. It's about assembling many sources to try to recover a picture of a world that we can never interrogate directly

You ask about many topics -- Jesus is mostly thought to have existed; while there's very little record of him outside of Christian sources, the description within Christian sources makes more sense if its referring to an actual person than invented. By contrast, when it comes to a figure like Moses, we can't say with any probability whether he actually existed, and there's no way to figure a claim like the story of his adoption by the Pharaoh's daughter. Did it happen? Or not? Or is it some shadow of another story, now forgotten? Short of a major discovery of some new material, we're not going to know. What we can say is that there are a number of other traditions in which "daughters of Pharaoh" are mentioned-- different stories, but why are they there?

Similarly, the story of Solomon and the United Kingdom period brings you to a very complex scholarly debate, it's a far more complex story than "the Bible is true" or "the Bible isn't true". Think of it as: the Bible is an accumulation of material that helps us get closer to the period, but like, say, the Iliad's memory of the Trojan War, we use it as a clue to what might have happened and where, not as proof that it did.

There's a complex battle over the First Temple period in Judaism. Much too complex to summarize here, but the short form is that you have to dig deep into the archaeological record to make sense of the Hebrew Bible, and even when you do there are plenty of question marks and competing interpretations. It's also important to note that this topic is more the domain of archaeologists than pure historians; we have so many archaeological materials -- and continue to find new ones all the time-- that archaeology has pride of place in this discipline.

The story of the Crucifixion itself is interesting, because it's by far the most detailed description that we have-- we know that the Romans did it, but it's otherwise mentioned only in passing.

So in approaching the Bible as an historical document, you need a _lot_ of other information to put any claim into an historical context. The faithful may read a sentence and say "it says so in Mark" -- and that will be sufficient for them; it never is for an historian.

Sources:

Cook, John Granger. “Crucifixion as Spectacle in Roman Campania.” Novum Testamentum, vol. 54, no. 1, 2012, pp. 68–100.

Assmann, Jan, Moses the Egyptian: The Memory of Egypt in Western Monotheism, Harvard University Press 1997

Steiner, Richard C. “Bittě-Yâ, Daughter of Pharaoh (1 Chr 4,18), and Bint(i)-ʿAnat, Daughter of Ramesses II.” Biblica, vol. 79, no. 3, 1998, pp. 394–408.

Finkelstein, Israel -- many works including The Quest for the Historical Israel: Debating Archeology and the History of Early Israel -- probably the most influential contemporary archaeologist on this topic.

Carr, E.H What is History? London 1961