The same way that they had until then. Only laws relating to the Holy Temple were moot.
They met and worshipped together on the Sabbath,They published many religious texts,They worked to convert the Babylonians to Judaism.
The Exile of Jews from palestine is known as the Diaspora
The Babylonians drove the Jews into their first exile. The Romans were the ones who caused the second and current exile period.
You're describing part of the diaspora (exile; galut), which is what happened when Jews were unable to continue living in the Holy Land.
It depends on the exile in question. Most Jews that wanted to returned to Land of Israel after the exile in Babylonia on account of Cyrus the Great. Many Jews have returned to Israel in the last 150 years from the Roman Exiling of the Jews nearly 2000 years ago. However, not all Jews have returned home after exile in both cases.
Galut; exile; diaspora.
Diaspora.
Cyrus
The Babylonians.
it ended in 1944
God was with them if they believed in him. In the modern era, most Jewish leaders believe that Jews are in Exile and subject to persecution explicitly because of the Jews' violation of this covenant.
The Jews who returned from exile in Babylon and their slaves, under the leadership of Ezra.