This was asked about a month ago over at /r/AcademicBiblical, and /u/SF2K01 links to another answer he gave over on /r/Judaism.

To quote a review of Schiffman and Cohen,

The real problem, though, is that there is simply not enough evidence to answer this question properly.

The general consensus seems to be that matrilineal descent began with or shortly after the Persian captivity. (Ezra, coming out of the Persian captivity, speaks of missing records and doesn't seem to be interested in only matrilineal nor only patrilineal descent.)

A point in favor for during or shortly after Persian captivity is Deuteronomy's prohibition against marrying foreign wives. (Deuteronomy was probably not written until the 5th or 4th c. BC. This is part of what's known as the Documentary Hypothesis. This thread is a decent introduction with links to further resources, or check out the /r/AcademicBiblical wiki.) Since the other books of the Pentateuch don't discuss marriage to a foreigner, it would seem this is a novel idea for the author of Deuteronomy. And indeed, you'll find the story of, for example, Ruth who is a foreigner that is actually part of the (legendary) lineage of David. This story is likely older than Deuteronomy, and so it is "great-great-grandfathered" in.

Because Jewish tradition traces their descendant from Abraham from his 2nd son Issac not his 1st Ishmael. Issac's mother was Sarah, Abraham's wife. Ishmael was the son of Haggar, her servant.