They are One God. But Jesus is not the Father and The Father is not the Son or the Holy Spirit. This is Trinitarian belief. Jesus has to be a different person from the Father to be sent by the Father (Gal 4:4). The Father has to be a different person from the Son to be the one who sends. Read the Cappadocian fathers Like Basil the Great or Anthanasius, they understand all this really well. Catholics honor these Eastern Fathers of the Church today.
Which is it? Unseparated, meaning mixed, and unmixed meaning separated.
Both, because He is one person, a hypostatic union. See the Council of Calcedon
https://www.ewtn.com/library/COUNCILS/CHALCHRI.HTM
They are essential because your Church teaches that Catholics believe in a false God.
You have to convince me first that you even understand what it means to believe in Jesus Christ the Son of God. Your questions indicate you've been unable to explain how His sacrifice on the Cross is efficacious for the salvation of sins.
Because really, if God did not act in the person of Jesus, therefore the salvation of humanity is untenable. But because He does, in the person of Jesus who is the incarnate God, He therefore saves. But the meaning of the incarnation literally means that Mary bore and gave birth to the Son of God. This therefore makes Him the Son of Mary And Mary the mother of God. Because such realities are communicable within in the second person of the Trinity. Its round and round we're going here...The incarnation is the essential reason for the title Mother of God. I'm attempting to explain it to you in many ways.
If you don't believe me, get a degree in Christian theology and come back and explain it otherwise.
This is pretty humorous; the Catholic Church, for some reason, does not believe that Christ's sacrifice on the cross is efficacious either; I do -- he paid the price for my sins; I'm saved by grace through faith. Catholics, on the other hand, are taught that you have to do more.... so, personally, I believe the issue here is what you, if you are Catholic, believe: is Christ's death enough? And, if so, why must works be added to this? Works are a natural outgrowth of salvation -- an evidence of it -- they do not aid in obtaining or securing it! If Christ's sacrificial death on the cross is enough, why do Catholics believe in Purgatory -- a final "purging'? Wasn't Christ's death and resurrection enough?
You can attempt to explain it many ways; I have an issue that the term implies that Mary bore GOD. She didn't. She bore Jesus. Jesus is God incarnate. God has no beginning and no end. I have no issue with professing the Trinity -- regardless of your belief, I am a Christian. However, it is the Catholic faith that raises Mary to a level that at best borders on idolatry and at worst IS idolatry. The CCC teaches that Mary is a "mediatrix" completely in defiance of Scripture, much less the very words of Christ himself! I have yet to see you explain this! The same entry indicates that Mary has a saving office. She does not. She cannot save; if she could have saved, there would have been no need for God to send his Son into this world, would he? Again, blasphemy taught as doctrine.
You are focused on the Mother of God issue so tightly that for some reason, you're not seeing why the sources I've listed for you that cause such consternation. CCC 969 is wrong; Mary does nothing to save, to mediate or to benefit. God's Word proves it to be a lie, yet you do not answer as to why it is not a lie. Why not? Does Mother Church trump God himself?
Hoss