I keep running into this misconception, and since time is precious I will write my thoughts here to save setting them down anew for each person I speak with about this. Kind of a work in progress since I will undoubtedly not get everything written first time around.
No. This comes from a muddling of terms. Modern Judaism has very little to do with with Ancient Judaism, the followers of which I will call Israelites to avoid confusion. The Israelites' religion was fulfilled by Jesus Christ, the Israelites who denied Christ stuck around and pretend to be the descendants of ancient Israel, wheras the Jews of the time who did not deny Christ were no longer called Jews, they were called "Christians". This is why modern Judaism is defined in negative terms, as a rejection of Christ, those Jews who do not reject Christ are no longer called Jews, but Christians.
This is nothing new in the history of Judaism however, since Jews are rather infamous for shunning and hating their own prophets even in their own religion.
When the Jews lost the temple, the religion lost its centre. Without a centre, it turned to Rabbinic Talmudism, and the Talmud became the new centre, superseded Mosaic law and Judaism became a religion ruled, rather than by the divine, by its own priests, with predictable results.
"Rabbi" as applied to Christ in the Bible was in the general sense of the word of "teacher" or "master". This is totally differing from modern, rabbinical Judaism that involves Rabbis contributing to the Talmud, a book that so harshly treats Christ it would be improper to quote some of the relevant passages here. This is another semantic confusion.
It would be like implying every time someone called a mentor "father" in the past, that meant they were actually Catholics since they call their priests "father".
No; Jesus was of a Levantine ethnicity you might call ethnically Jewish, but this would be another semantic mix-up. In modern times "Jew" refers to someone who is religiously Jewish i.e. rejects Christ. We'd typically make it clear its ethnicity we're talking about either by context or being more specific i.e. "Ashkenazi Jew", "Shepardic Jew", and as such Jesus was not a Jew.
In the same manner, it would be incorrect to call Moses, Abraham or any of the Old Testament figures "Jews", since the group of people we call Jews did not come into existence until Christ, as before his life there was no Christ to even reject. Those people are better called Israelites.