There is no archaelogical evidence for Kings David or Solomon, despite the feverish efforts of Israeli archaeologists over the past 70 years. It is likely that they are fictitious. There is good reason to think that the Babylonian captivity happened, at least in some way, as the Persians were in the habit of forcibly relocating conquered peoples. Carvings on some persian palaces show this. Some people think that the jews were canaanites who were forcibly removed for a time to the Iraq area, and later allowed to return, having been influenced by persian customs. They were seen as outsiders by the canaanites who had remained. Whatever happened, there is no doubt that jews were present at the time of the roman conquest (there wasn't much in the way of an independent israel, as it was variously controlled by the Hellenistic successor states), and the romans did indeed destroy the temple that was there round about that time. There is ample evidence of this--the arch of Septimus Severus in Rome, for example. Whether the temple was the first, second, third, or whatever temple is unknown. Likely such a prominent hill would have been considered special by the inhabitants, whoever they were. For that reason, even if archaeology were allowed on the Temple Mount, (it isn't) simply finding older buildings than the Roman-era temple would not be 'evidence' of the existence of previous temples, unless they contained recognizably Jewish artifacts or writing.
I've heard it said that the western wall itself was part of a later arabian fortress, but I don't have a source for that.
