Whatever religion that a person believes in is completely up to that individual, and no one can say whether their belief is false or correct. Religion is simply believing in a higher power that controls parts of life that you can not explain, so believe what ever you want to.
There is no reason why a Jew cannot believe that a man called Jesus walked the earth 2,000 years ago. The only thing a Jew would not believe is that Jesus was the second coming of God. A Jew can certainly believe that Jesus was an important prophet, just not the Messiah.
I believe someone who follows Judaism can believe in Christ as a many and still practice their faith. It is in admitting that Jesus is the son of God or the King of Jews that makes the waters a little unclear. Christianity was started by Jewish people who believe in Jesus, but they were effectively no longer Jewish at that point.
Yes, a person can still believe in Christ while practicing Judaism, because there is nothing about the tenants of Judaism that reject Christ as the Messiah. Christ is the key part of Judaism, and many Jews recognize that He was the true Messiah. That Christ came is also not a reason to throw out the Old Testament, so many Jews strike the correct balance.
They refer to themselves as fulfilled Jews. There is nothing about being Christian that would prevent someone from celebrating the Jewish feasts and holidays. To worship God is to worship God.
So long as they practice their Judaism in a way that is not in conflict with Christianity, I don't believe that any Christian would criticize either.
A person who believes in Christ would be a Christian. Although there is a branch of Judaism that calls itself Messianic Judaism that seems to think otherwise. I'm not sure if they're the same thing as Jews for Jesus, who also have similar beliefs. They're not taken seriously by mainstream Jews.