Were the eirev rav Jewish? If they were in fact Jewish why are they described by the term eirev rav instead of just referring to them as we would any other Jew? If they were not Jewish why were they not cast out after repeatedly causing trouble?
Rashi in Parshas Bo 12:38 says that it was a mixture of converts that went out together with the Jews. Perhaps it was difficult to cast them out as there were bona fide Jews that were also part of the Erev Rav.
They stood at Mount Sinai when the Torah was given, so I would think that it was given to them too, as they were then part of the nation. Also, if they were Egyptians, they did not drown in the sea - so they must have become Jewish. Perhaps accepting the Torah allowed them to become Jewish. And if they were not Jewish, then they likely died during the generation of the desert because they would also have spoken badly about the land. Interesting question though.
They were from the Jews that Hashem wanted to kill at the plague of darkness so Moshe begged Hashem to give them a chance and this is what happened