I understand that being a Jew is both a religion and a race. Can you be a Christian Jew? A Muslim Jew?

No

Being a Jew is not a race.

Being a Jew is to be a member of a nation.

The Jewish religion is the law of the Jewish nation.

To become a member of a foreign religion (Christian, Muslim, Buddhist)

a person would be affirmatively rejecting the principles and laws and ...heritage of the Jewish nation.

This is rejection of citizenship in the Jewish nation.

Thus - such a person is treated for all purposes of Jewish national belonging (marriage, burial, holding office in the Jewish tribal / religious structures ...etc.) as a foreign person.

However, as he started out as part of the Jews, and therefore subject to the "Jewish covenant with God" -- the religious belief is that he will be subject to whatever divine punishments come from breaking the covenant.

A person who left the Jewish people fully voluntary is considered essentially "a betrayer"

While a person who did so involuntary is considered "captive among the foreign nations"

(for example - when Jews were forced into Christianity or Islam by torture and threats)

For drastic examples of these two --

Torquemada (one of the Grand Inquisitors of Spain) was born a Jew, converted to Christianity voluntarily and then began to persecute Jews, create false propaganda to vilify them, torture them...etc.

So - he would be a "betrayer"

Cardinal Lustiger was born a Jew.

As a child, he was hidden from the Nazis by a Catholic couple. He was brought up in Catholicism and his theological concepts were all shaped through it.

Thus, he accepted it at a young age, and eventually raised through the ranks of the Church.

Nevertheless, he retained affection for the Jewish nation.

This is a case of a person who did not really make a choice of rejection -- but was fitted to it through circumstances.

He was respected by the Jewish community, made welcome, and went into Synagogues to say Jewish prayers for his parents.

He is considered "a captive of the nations"

Yes you can.

quran 2:135

They say, "Be Jews or Christians [so] you will be guided." Say, "Rather, [we follow] the religion of Abraham, inclining toward truth, and he was not of the polytheists."

jesus and ibrahim were muslims nor jew or christian

No to Judaism being a race. Jews come in all races. All you need is a Jewish mother. Her race doesn't matter.

No to being a Christian Jew. Judaism does not accept Christ as God & Christianity does. Ditto for Islam.

Nope, A Christian is a Christian !!

If it's a religion, and you understand that, how can you be a Muslim or Christian Jew when those are both religions? It would be like being a Christian Muslim, silly.

No, we are not a race.

And no, one cannot be a Christian Jew or a Muslim Jew. If a Jew starts observing other religions, he is no longer Jewish.

Judaism is not a race at all. There are Jews of all races. In reality, there's only one human race, but there are Jews who belong to all three of the traditional racial categories. The first Jews were just people of the Mideast like many others who lived there. They didn't belong to some separate "race". The term "Semitic" refers to a language group, not a race.

It's a religion and an ethnicity, but there are actually more than one Jewish ethnicity. Ashkenazim and Sephardim represent the two major divisions.

Many Jews have converted to Christianity. In fact, the first Christians were Jewish-born. Such people could be called "Jewish Christians," but that term is not normally used.

Over time, many Jews in the Middle East had reason to convert to Islam. The Palestinians are partly the descendants of such people. You could call such converts Jewish Muslims, if you wanted, but again, no one uses that term.

There's no reason to do so.

My friend converted to Judaism to marry. She was English, more or less, Christion. The children are growing up as Jews but the family celebrates every Christian and Judaic festival. She says she's never spent so much time taking down and putting up stuff.