The Adam and Eve story was a myth that tried to explain how death, toil and other woes came into the world; and like the Greek Pandora myth, a woman is blamed (and by extension, in a patriarchal society, all women). The pain of childbirth, and menstruation, is explained in the story when God curses the woman - shortly after called Eve by Adam - as you have quoted.
The first creation story in Genesis 1 seems to imply that men and women were created at the same time, and men or both men and women were created in God's image depending on what translation you prefer (verses 27 to 29). In the second version, starting in the 2nd chapter, the first woman is made from Adam's rib, so in this version, man came before woman.
The creation of humanity is central to the second creation story in Genesis. In this older narrative the first woman is created after the first man (Adam), from his body (Genesis 2:21-22), a situation which no doubt reflects the dominant position of males in a patriarchal society.
But the implied inferior status of women becomes explicit only after the fall from grace in Eden, when God tells the woman: "your desire shall be for your husband, and he shall rule over you" (Genesis 3:16).
Adam then gives his partner a name, Eve, an act that echoes his naming of the beasts, which expressed his dominion over animalkind.
However, this account stresses, even more strongly than the first story in Genesis, the unity of the sexes and their mutual need. God says: "It is not good that man shall be alone; I will make him a helper and a partner" (Genesis 2:18). Only woman proves suitable for this role.
Paul believed that "Adam was first formed, then Eve" (1 Timothy 2:13). And he also states that "... he (man) is the image and glory of God: but the woman is the glory of the man. For the man is not of the woman: but the woman of the man" (1 Corinthians 11:7-8).
Modern concepts imply that men and women evolved together, there already being male and female animals from ancestor species since well before the species that was the common ancestor of all primates.
Except for the 1st, 2nd, penultimate and last paragraphs: J R Porter, Professor Emeritus of Theology, University of Exeter: The Illustrated Guide To The Bible