Did Jesus Christ claim to be God?

Yes. At his trial the Jews recognized that Jesus was claiming to be the son of God the Father, and they condemned Him to death precisely for making that claim. See Matt. 26:62-66. On various occasions too, Jesus claimed to be God. For example, he had announced to a paralysed man that his sins were forgiven. The Pharisees, who heard him, thought in their hearts, “Who is this who speaketh blasphemies? Who can forgive sins but God alone?” (luke 5:21). Jesus, giving expression to their thoughts, worked a miracle that all could see, to prove that He could, indeed, forgive sins, i.e., that He was God. Jesus even applied to Himself the very title for the Almighty,” I AM”. “The Jews therefore said to him: Thou art not yet fifty years old; and hast thou seen Abraham? Jesus said to them: Amen, amen, I say to you, before Abraham was made, I AM.” (John 8:57-58)

Is it not possible that Jesus, when he said He was the Son of God, was merely claiming to be a prophet sent by God?

No. In the first place such a claim would not have been taken as blasphemy. Moreover, Jesus called God “My Father”, in a way in which no mere man could speak of God. On one occasion He said openly, “I and the Father are one”. The jews took up stones to cast at Him, saying, “Thou being a man, makest thyself God.” (John 10:33)

Does not Jesus at times speak as if He were inferior to the Father and at other times as if He was equal?

Yes. For example, He said, ” The Father is greater than I” (John 14:28) and also “All things whatsoever the Father hath are mine.” (John 16:15) Jesus Christ is one Person. But that one Person is both God and Man. At the time of the Incarnation He took a human body and soul. As God, the second divine Person, He always existed. The second divine Person, therefore, now possesses two natures, a human and a divine. In the Gospels He acts and speaks sometimes in His human nature i.e. as Man; sometimes in his divine nature, i.e. as God. He spoke in His human nature when he said, “The Father is greater than I”; and in His divine nature when he said, “All things whatsoever the Father hath are mine.”