The Relationship Between Judaism and Christianity

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And I'll do the same for a quick summation -

 
Was Jesus a Jew?
 
According to the Christian bible he was and if he was born to a Jewish mother, then yes, according to Judaism he was as well.
 
Were the followers of Jesus also Jews?
 
I'll agree with Channalee here, probably some of them were, and some of them weren't.
 
Was Jesus the Messiah?
 
Certainly not the Jewish messiah as we can see the world has not changed as the Hebrew bible explains it will when the awaited messiah arrives.
 
Do Jews cease to be Jews the moment they take on the belief that Jesus is the Messiah?
 
I'll also agree with Channalee here and add they will still be Jews but they are not practicing Judaism, which are 2 different things.
 
Is Christianity a "branch off the tree"?
 
Nope, Christians aren't "grafted" in as the Christian bible would have you believe. You will see throughout the Hebrew bible that G-d makes a clear distinction between the Jews and the Gentiles. Not that the Jews are better, they just have a different mission.

Hope that helps to answer some of your questions!