Yes, Adam's wife's name was Eve. Adam did not have a mother, he was the first human being created. Eve was made afterwards, from his rip, the second human being created. from these two, all humanity has descended.
You seem to be caught up on the king james version of the bible which says that God called Adam and Eve by one name: "Adam". Although I have no idea what it is you think that signifies, I would encourage you to look into other translations of the bible. The vast majority of them have that verse as "and called their name 'mankind'", or "and called their name "human beings".
I am sorry for being rude and insensitive to you, i know you are probably just doing your best to try to seek God in His word, truth from the bible. But at the same time you really ought to do your research, better yet, find a good church and talk to the Pastor and get in some small groups and learn this stuff from other Christians who have been walking with the Lord for a while.
You will find that traditional Christianity has a very concrete foundation for its main doctrines; Jesus Christ.