Well, there's certainly nothing that prohibits a few "strays" from any of the tribes ending up somewhere. As you can see from the population numbers at my Profile, we are talking LOTS and LOTS of people, especially from the Northern Kingdom.
Since only ~50,000 Judeans returned from Babylon, that leaves another half million unaccounted for. Technically they are not Jews since only those who returned from Babylon were accorded that name, but they certainly were Judeans, and essentially the same people. They also knew and practised the tradition (we assume), even though they didn't return. Maybe some of them ended up in India?
That cannot be said for the Northern Kingdom taken into Assyrian captivity. They had lost virtually all contact with their religious and cultural roots even BEFORE their captivity. That, along with their diaspora was part of their punishment. They carried little or none of it into captivity or beyond. That's why its such a big error to attempt to find the Lost 10 Tribes of Israel practising any ancient Israelite customs. And that's why obscure little tribes found here and there, practising ancient Israelite customs or claiming to be The Lost Tribes (for whatever reason) are most certainly not.
This is absolute nonsense. There is no difference in the Hebrew language between the words "Jew," "Judean" and "Tribe of Judah." They are the same thing.
The large number of Jews who remained in Babylonia, rather than return to Israel in the time of Ezra, are not unaccounted for, as you claim, but are well accounted for and were always fully accepted as Jews, as were the equally large number who had fled to Egypt and chose to remain.