You must mean the first temple. The second temple was the one established in the Persian period and rebuilt by Herod.

There's not much to say about Jerusalem after the Babylonian invasion and destruction of the city. There is almost no evidence of habitation during the ensuing period. It was either abandoned for a time or inhabited by a very small population (maybe a few hundred).

It was probably the Persians who later helped the Jews reestablish the Jerusalem temple and priesthood. Jewish practices also continued at other locations, e.g. the temple in Elephantine.

This is not an /r/AskHistorians calibre answer, but since the thread has gone dormant I'll take a stab at it to at least point you in the right direction.

My amateur understanding of the climate of the Babylonian destruction is that the Persians, who later conquered the area, had a polar opposite approach to their controlled areas. Whereas the Babylonians exiled the ruling classes, the Persians had a policy of having friendly alliances with local rulers. They had the idea that there would be less unrest if the local people could govern themselves to some extent. The Cyrus Cylinder was a decree that allowed for the repatriation of the Jewish people.

Going forward to the destruction of 70AD, my knowledge is even shakier, so I can only offer that the region continued to have violent protests to Roman rule. The most well-known of these uprisings were the Siege of Masada and the Bar Kokhba Revolt in 132AD, which led to the expelling of the remaining Jewish population.

I hope this at least gives you something to go on as you explore this fascinating topic!