The "man giving birth" to woman story is actually a story that shows the relfective nature of the union between man and woman of the union of the Trinity. This is why Adam says "bone of my bone, flesh of my flesh."
The Trinitarian union goes something like this: God the Father begets God the Son. God the Son is begotten of the Father, because He is the product of God's perfect knowledge of Himself made real. Thus, since one is the perfect knowledge of the other, they are identical, yet fulfill different roles, based on the nature of the relationship between them. The Holy Spirit is the perfect love between the Father and the Son, which is intellectually-based (ie, the Son knows the Father perfectly, and loves Him for who and what He is, and vice-versa), but the Holy Spirit is also the product of that love, and is often referred to as Life. The statement by Jesus: I Am, the Way, the Truth and the Life, refers to this inner reality of the Trinity. The Father, being the origin, the Creator, could be identified as Will, or Way. The Son, being the perfect knowledge of the Father, who is also referred to as the Word in the Gospel (particularly of John), could be identified as Truth. I have already shown that the Holy Spirit is identified with Life.
Thus, the relationship between the man and the woman is said to resemble this inner nature of God. God the Father, being the initiator, begets the Son out of His own being. The man, being the initiator, has the woman 'begotten' (if you will) from his own flesh. "Flesh of my flesh, bone of my bone." They are one as the Father and Son are one. The child is the fruit of the love that exists between the husband and wife, just as the Holy Spirit is the fruit of the love which exists between the Father and the Son.
Thus, one may be left wondering, why do we refer to the Son as such, why not as wife? It is for different reasons. One reason being that the Son is still God, and as such, is still initiator of the created world. In fact, when mankind was created, God said, let us create man in our own image (here referring to human intellect, free will and capacity to love - love being intellectually based here). Thus, together, the Trinity is the Creator, and not simply one specific Person (though creation is often identified with the Father). Another reason being that the Son was begotten of the Father, proceeding from His being.
The role of the Second Person of the Trinity is the union of God with His creation. Prior to the "Fall" of man, the Second Person still performed this role. However, sin is the dividing force between man and God, since sin is the choice (well, certain sins, there are many sins which do not to this, but are simply bad tendencies and habits, faults, etc.) to willingly separate one's self from God. Thus, in order to regain the unity of God to man and man to God, the Second Person came, as man, to be a sacrifice on behalf of man, to unite Himself to all of man (through Baptism, in which the life of the Holy Spirit is once again born in man, which was man's natural state prior to sin... through Baptism, but more fully through the Eucharist, whereby we are made one not merely with Christ, Body, Blood, Soul and Divinity, but also with His sacrifice, making it a sacrifice, not of one man, but of all mankind, through Him). Thus, reinstating the union between man and God, which is what the true state of heaven is.
Creation was the union between God, the all-actual, and pure matter (as Aristotle envisioned it), the all-potential. Or, the all and nothing. The created universe was the product of that union. Thus, in Genesis, is said the "Spirit of God moved across the void." Thus, just as a child is born of two parents, the created universe is the "child" born of God and matter. Therefore, we see that the created world is both actual and potential, a combination of the two, whereby the potential actualizes, yet potential still exists, being infinite. This is the nature of cause and effect. What is potential comes into union with the actual, and the potential is actualized. The actual is the cause. The potential is what is acted upon to produce the effect of actualization of something new. Union between the initiator, or the actual, with the receiver, the potential, to produce something new, a new actual.
The Second Person of the Trinity is referred to as Son, because of the nature of His relationship to man, the unifying force between God and man. Thus, the nature of the Second person is intimately tied to creatures, the children of God.
Just like the Holy Spirit is the product of the union between the Son and the Father, the Holy Spirit remains that product of that same union in man. The nature of the Son is intimately tied to mankind, thus, the love of the Father to the Son, and vice versa, is intimately tied to the love the Father to His creation, thus, the Holy Spirit is both the Divine love between the Father and the Son, but also the Divine Life in us that gives us spiritual life whenever we are in union with God.
All things are about union. The union between the initiator and the receiver, and the product of that union, which is life out of love. The relationship between a man and a woman is a relfection of that unitive reality. Thus, we see that Genesis shows the reflection of God's inner union, being God from God, Light from Light, True God from True God, as woman is from man, "flesh of my flesh, bone of my bone."