Eve was created as Adam's wife.
In Genesis 2:24-25, speaking of Adam and Eve immediately after the special creation of Eve, we read "For this reason a man shall leave his father and his mother, and be joined to his wife; and they shall become one flesh. And the man and his wife were both naked and were not ashamed."
The Hebrew word for "wife" is "'ishshah", which can also be found in the following passages:
Besides, she actually is my sister, the daughter of my father, but not the daughter of my mother, and she became my wife ('ishshah)...
Then Isaac brought her into his mother Sarah's tent, and he took Rebekah, and she became his wife ('ishshah), and he loved her; thus Isaac was comforted after his mother's death.
So Boaz took Ruth, and she became his wife ('ishshah), and he went in to her. And the LORD enabled her to conceive, and she gave birth to a son.
The word "'ishshah" can also simply mean "woman", as in the following verse, describing an unwed virgin:
"The servant said to him, 'Suppose the woman ('ishshah) is not willing to follow me to this land; should I take your son back to the land from where you came?'"
If the word "'ishshah" can refer to a wife, or to an unwed virgin, how are we to know what it is talking about in Genesis 2:24-25, when it is used to describe Eve?
We can be certain that Genesis 2:24-25 is not speaking of Eve as a generic woman who is not joined to Adam because of the context and grammar of the passage. In the passage itself, she clearly belongs to Adam (she is HIS woman), and their joining together is described as the model for all future marriage.
Further, Jesus quotes this passage in Genesis in Matthew 19:3-6 when asked about divorce, and His words again make clear that the context of Genesis 2:24-25 is that of marriage, and not simply that of some man and some woman to whom he is not married.
Having said all of that, there is no reason to believe that everyone in the Bible who has children must have been married, simply because Christians believe that sex should be confined to marriage. There are numerous examples of non-marital sexual relations in the Bible, some of which end in the births of children (Genesis 19:36).
May God bless you as you study His word.