Is there an appropriate name for our Savior in the context of certain situations? Should we refer to him as Jesus, Christ, Jesus Christ, Christ Jesus etc.?

As your question suggests, exactly how we address Jesus will vary in certain contexts. Perhaps it is most important to distinguish at the outset that “Christ,” is not part of Jesus’ proper name, but rather, it is his title. In this sense, it sometimes helps to put the definite article “the” in front of Christ, saying, “Jesus the Christ,” to remind us that “Christ” is not his surname. Jesus is “the Christ,” Which means, "the anointed one" and translates the Hebrew word “Messiah.” 
 
His proper name, "Jesus" means, in Hebrew, “God saves.” To put all of this together in English, his name and title might be rendered “Anointed Lord and Savior.” 
 
While it is certainly fine for us to call him simply by his proper name, “Jesus,” in the formal liturgy of the Church we often speak of him more fully, such as, “Christ our Lord,” or “Our Lord Jesus Christ,” and so forth. 
 
All this said, it must be added that there are over 100 titles of Jesus, and ways of referring to him in the Scriptures which can also be appropriate ways of referring to him in certain circumstances. For example he is called: Alpha and Omega, Author and Finisher of our faith, Son of David, Son of Man, Good Shepherd, Emanuel, I AM, King of Israel, the Way the Truth and the Life, Light of the World, Redeemer, Teacher, Rabbi, Son of God, Son of Mary, True Vine, and so forth. There are more than 100 other such titles. 
 
Thus, we do well to remember that the magnificent truth of Jesus Christ, our Lord, often requires us to speak of him in many ways, pondering his glory from many different perspectives through these titles.