Hebrew did fall out of spoken use by the time the Jews were exiled into Babylon. However existing prayers and of course the Torah-Prophets-Writings itself was studied in the original Hebrew. And with the exception of Babylonian writings, Jewish scholars almost always wrote their works in Hebrew.

More recently (200 years ago) some prayers were translated and scholarly writings written in German, to appeal to those who were more assimilated into German society. Also at some point Yiddish (a combination of Hebrew and German) became a language of use among Jews in Europe. And at that time a major effort was made to revive Hebrew as an actual spoken language, with a couple of newspapers and study groups popping up in parts of Europe.

In the meantime, Middle Eastern Jews were speaking Arabic. My guess is there were a handful Middle Eastern Jews in Israel (or whatever it was called then) when the Turks were in power, so they were speaking Arabic (and praying and studying in Hebrew). When the European Jews returned to Israel - which started in the 1800's - they brought in the languages of their home countries (Russian & Polish much more than German), Yiddish, and a revived spoken Hebrew.

I don't think Latin was used by Jews very much at any given time - the Septuagint was a translation at the request (or command) of the authorities at that time.