Moses instituted the Mosaic Law well before the Captivity event.
Prior to that, you had the Israelite tribes, the ancestors of the Jews to be sure, but it’s a good deal fuzzier. There’s no sharp dividing line, so a First Century Jew would have been correct to see themselves as a descendant of the Israelites, but it’s a bit like how a Frenchman might consider himself a Gaul, though obviously a millennia and a half separate the modern French resident from his Gaulish ancestors.
No. In fact, they kept strict family records. They knew who was descended from whom. If you read the books of Kings, you will find the understanding you seek there.
The dividing line is the death of Solomon. His sons angered the other tribes and they broke with Jerusalem. That is when they set up their altar on the mountain and began to worship God outside of Jerusalem. Then came the captivity and then the remnant was returned to Jerusalem and they rebuilt and then came Jesus. And He said,
Matthew 23:2 Saying The scribes and the Pharisees sit in Moses’ seat:
Thus recognizing that throughout the whole ordeal, the Jews had continued to worship according to the Mosaic Law.