What a great question!  The Bible is not silent on this matter, so let us begin by reading John 1:1, “In the beginning was the Word, and the Word was with God, and the Word was God.”  “In the beginning” gives the thought of a time when anything that had a beginning actually began.  Of course, God is eternal, so when time began, He was already there as THE WORD.  A word is the expression or utterance of a thought.  As THE WORD, God is the manifestation or expression of Himself. 

Now let us read John 1:14, “And the Word was made flesh, and dwelt among us, (and we beheld his glory, the glory as of the only begotten of the Father,) full of grace and truth.”  This is speaking of the Lord Jesus Christ.  As God, He existed in eternity past.  At one point, He ‘was made flesh.’  In other words, He was born.  God became flesh and blood.  God became something He had never been before.  He was born into the word and took on flesh and blood.  He became a man.  However, He did not cease being what He always was…GOD.  Jesus Christ is God in the flesh!  In the Lord Jesus, “God was manifest (made known) in the flesh…” (1 Timothy 3:16). 

The Lord Jesus, Himself declared that He was God while He was on the earth.  In John 10:27-32 He said, “My sheep hear my voice, and I know them, and they follow me: And I give unto them eternal life; and they shall never perish, neither shall any man pluck them out of my hand. My Father, which gave them me, is greater than all; and no man is able to pluck them out of my Father’s hand. I and my Father are one. Then the Jews took up stones again to stone him. Jesus answered them, Many good works have I shewed you from my Father; for which of those works do ye stone me?”  Here, the Lord Jesus claims equality with God, the Father.  Upon this claim, the Jews sought to kill Him.  Why?  We read in verse 33, “The Jews answered him, saying, For a good work we stone thee not; but for blasphemy; and because that thou, being a man, MAKEST THYSELF GOD.” These people understood that in His words, the Lord Jesus claimed to be God. 

Only God deserves to be worshipped.  We see an angel of the Lord telling the Apostle John, “Worship God…” (Revelation 22:9).  Yet, during His time on the earth we find the Lord Jesus accepting worship from others.  He was worshipped by a man whom the Lord had healed of blindness.  “And he said, Lord, I believe. And he worshipped him.” (John 9:38).  The baby Jesus was worshipped by the wise men (Matthew 2:11).  Mary Magdalene worshipped the Lord (Matthew 28:9).  There are other incidents where the Lord Jesus was worshipped and received worship, thus manifesting that He is God…who alone greatly deserves worship.

During His life on the earth, the Lord Jesus manifested the characteristics of God.  He was sinless (1 Peter 2:21-22).  He was omnipotent (Luke 8:25).  He was self-existent (John 1:3).  The Lord Jesus showed that He was God in His words and in His actions.  John said of the Lord Jesus in John 21:25, “And there are also many other things which Jesus did, the which, if they should be written every one, I suppose that even the world itself could not contain the books that should be written. Amen.”  There are many other proofs to show that the Lord Jesus was fully man and fully God while on the earth.

You asked if the name Jesus was given to Him when He became a man.  Yes.  Although He has many names (Jehovah, El, El Shaddai, etc.), He was given the name Jesus at His birth.  The angel of the Lord said to Joseph in Matthew 1:21, “And she shall bring forth a son, and thou shalt call his name JESUS: for he shall save his people from their sins.”  The preciousness of that name will last for eternity.  We read in Philippians 2:10-11, “That at the name of Jesus every knee should bow, of things in heaven, and things in earth, and things under the earth; And that every tongue should confess that Jesus Christ is Lord, to the glory of God the Father.”  For eternity, that precious name will move us to worship and adore the One who loved us and gave Himself for us.  (465.6)