I can answer the question of what we know about David and Solomon very simply. Archaeologically we know absolutely nothing. Historically we know virtually nothing. All we know about them are stories about them and the stories are all in the Bible. So, the question is can we convert Bible knowledge, stories about them into some other kind of knowledge.
Directly, we can’t. The only way we can do it indirectly is to find archaeologically or historically some kind of other evidence that makes it plausible for these characters to have existed, though even plausibility doesn’t guarantee anything. But that would be a step forward. Unfortunately on that I have to say also that the archaeological evidence is generally speaking against, rather than for. I think for it to be historically likely that Solomon and David existed, the archaeological evidence would have to be predominately with, given that this is indirect evidence anyway.
So when people say, we have found a Davidic building or something, what they really mean is we found one from the time that David existed, which is completely circular, because we don’t know that he existed so that doesn’t really prove anything.
David and Solomon, however, were slightly different cases. I think that the tradition about David may have a historical basis; and he’s not possibly a completely invented character, only mostly invented. I think Solomon is a totally invented character, but the reasons are very technical. It’s to do with the nature of the Biblical material. There were little things like the fact that he is, his reign is described in a way that the others are not. He is only crowned once, whereas, his predecessor and his successor were crowned twice. So there is some historical reminiscence of there being two separate kingdoms there, which is not the case for Solomon. He rules over an empire, a kingdom that is slightly different from the kingdom that David is said to have ruled over, or Rehoboam. But there are lots of other little cases like that where I think Solomon is even more fictitious. But I will have to say now, I think, David in the sense that most people understand David, is a fictitious character and we have no historical information that would lead us to him in any way.