Someone who is born of a Jewish mother, or who has undergone a halakhic conversion.
Only one was called Judah, the King line.
To be technical, yes. But the name "Jew" has been used generically to refer to all identifiable "Hebrews" for a long, long time.
We are told that the House of Israel was sent into captivity and eventually to the four corners
Yes, but there were members of the 10 northern tribes who fled into Judah as well. So present-day Jews are really descended from all the original tribes, not just the tribe of Judah.
There have been a few cases of "lost tribes" who have sought to return to a connection with Judaism. Typically they have had their status "normalized" by conversion.
This may be what you are taught but reading of the OLD Testament Prophets says otherwise.