Was the Holy Spirit or God (the Godhead), the father of Jesus? Holy Spirit conceives Jesus

Matthew 1:20 ESV


  But as he considered these things, behold, an angel of the Lord
  appeared to him in a dream, saying, “Joseph, son of David, do not fear
  to take Mary as your wife, for that which is conceived in her is from
  the Holy Spirit. She will bear a son, and you shall call his name
  Jesus, for he will save his people from their sins.”


Luke 1:35 ESV


  And the angel answered her, “The Holy Spirit will come upon you, and
  the power of the Most High will overshadow you; therefore the child to
  be born[a] will be called holy— the Son of God.




Jesus is distinct from God

Clearly during "life" Jesus was distinct from God:

John 20:17 ESV


  Jesus said to her, “Do not cling to me, for I have not yet ascended to
  the Father; but go to my brothers and say to them, ‘I am ascending to
  my Father and your Father, to my God and your God.’”


Matthew 27:46 ESV


  And about the ninth hour Jesus cried out with a loud voice, saying,
  “Eli, Eli, lema sabachthani?” that is, “My God, my God, why have you
  forsaken me?”


And in heaven:

Psalm 110:1 YLT


  A Psalm of David. The affirmation of Jehovah (LORD) to my Lord: `Sit at My
  right hand, Till I make thine enemies thy footstool.'


and

Mark 16:19 ESV


  So then the Lord Jesus, after he had spoken to them, was taken up into
  heaven and sat down at the right hand of God.




The question

So then, the Holy Spirit must also be distinct.  Is it better proper to say that the Holy Spirit is the father of Jesus?
The Holy Spirit is not the Father of Jesus, God the Father is. What you are asking about is the nature of the Trinity.

Trinity

Christians have always believed the Son of God was preexisting before the creation of the world and is God who created the world. The Father spoke through the Son and the Spirit performed the actual creative events.


  John 1:1-3
  1 In the beginning was the Word, and the Word was with God, and the Word was God. 2 He was with God in the beginning. 3 Through him all things were made; without him nothing was made that has been made.


God told Moses that His name was ‘I AM’ (Exodus 3:14 ) meaning without beginning and eternal, therefore, when Jesus said the following, the Jews picked up stones to stone Him for blasphemy:


  John 8:57-58
  “You are not yet fifty years old,” they said to him, “and you have seen Abraham!” 58 “Very truly I tell you,” Jesus answered, “before Abraham was born, I am!”


The Holy Spirit is also God. He is an eternally existing divine person and the author of divine operations.

For example:


  Genesis 1:2
  Now the earth was formless and empty, darkness was over the surface of the deep, and the Spirit of God was hovering over the waters.


Now all three person of the trinity combined are the Godhead who is One.


  Matthew 28:19
  Therefore go and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit.


God is One:


  Isaiah 44:6
  The Lord, Not Idols 6 “This is what the Lord says— Israel’s King and Redeemer, the Lord Almighty: I am the first and I am the last; apart from me there is no God.


So it seems very Biblical to accept the Trinity as believed by the Christian church throughout all the ages.  One definition of the trinity that makes things clear can be taken from John Owen’s Works Volume 2, Page 462:


  God is one; — that this one God is Father, Son, and Holy Ghost; — that the Father is the Father of the Son; and the Son, the Son of the Father; and the Holy Ghost, the Spirit of the Father and the Son; and that, in respect of this their mutual relation, they are distinct from each other.


What sometimes makes the issue more complex, and more direcct to your point, is that Jesus sometimes speaks from His 'God-ward' side, like when the Jews wanted to stone Him, and sometimes from His 'man-ward' side.  The verses you quote seem to be when He is speaking from his man-ward aspect, like ‘to my God and your God’. The same is true when Jesus refers to his disciples as his 'brothers'.  This is easily understood when we remember although Christ was eternal, the man Jesus was not. Therefore, the Spirit is one person of the Holy Trinity performing the works of the eternal Father’s will through the eternal Son, God's word. That is how in a theoretical-sense the Spirit is kind of a 'father' of the body of Christ, but not really, more like the creative cause, or operation, according to the original Father's will. The God-Man is by no means the Son of the Spirit, He is the Son of God the Father.

The Son is the eternally generated Son of the Father, not the Spirit.  What you have noticed is that the body of Christ was made by the Spirit, so technically, yes, God was the father of the Man Jesus, by the Spirit’s operation and power. The eternal Son was willing to take this body prepared for Him, thereby creating the God-Man, the Christ of God, in order to bring salvation to men. By this means The eternal Son assumed a human body prepared for Him by the eternal Spirit, according to the will of the eternal Father.  All three; Father, Son and Holy Spirit, are the 'I AM', the only One and true God.