In a relational sense. Besides if you read Matthew 1, you do NOT see female god and male god having intercourse in any sense of the word. Most likely, the virigin birth was a special creation in the womb of Mary employing DNA from Mary and possibly from Joseph too. This last part is specualtion on my part.
NO! Joseph had no biological ties to Jesus! That is totally against Catholic Teaching!
Now, as far as Christ being the son of God. Because as Jesus he was the son of God, it was the union of God with Mary that produced Jesus. The trinity is, I think, the most difficult part of Catholicism to understand.
Let me see if I can explain. God the Father is called the creator. It is he who is most responsible for the creation of the world and the angels. But at the same time, since there is only one God, the Father, the Son and the Holy Ghost were all equally responsible for creation. Jesus is called the redeemer because he is the one who suffered, died and rose again. But at the same time, God the Father, God the Son, and God the Holy Ghost suffered, died and rose equally. God the Holy Ghost is the one who inspired the apostles and through whom Our Lady became the Virgin Mother of God. But, the Father, Son and Holy Ghost all had an equal part in this.
Our Lady was the mother of Jesus, but as the mother of God she was equally the mother of God the Father, Son and Holy Ghost.
I think it is just that it is easier for us as human beings to separate the three. Since the first person of the trinity, sort of “fathered” creation, he is called God the Father. Since the second person of the trinity, was the son of a woman and God, he is called God the Son. And since we know so little of the third person of the trinity, since he appears only in a breath of wind (almost as a ghost or spirit) and since he inspires, he is called God the Holy Ghost.
So, you see, the names are not perfect, it is just the best way we as humans have of explaining the separate natures of the three persons of the trinity, but we must never forget that at the same time they are one God with tree divine persons.
I hope that helps.