As sister *charisma* said, when we believe in Islam we believe in the whole of it, and don't and can't pick and choose. If we accept the Qur'an as the final scripture, and as being 100% the word of God, then we must believe in all the prophets, and belief in them all is one of the articles of faith. Indeed, in the Qur'an, Allah chastises the Jews for rejecting certain of the prophets and messengers He sent. Rejection of or blaspheming of even one of the Prophets takes one outside the fold of Islam.
Yes, we believe that Adam (peace be upon him) was the first human being, and Eve (Hawwaa in Arabic, peace be upon her) was his wife. They are the parents of the human race, and are our first mother and father. Allah chose Adam, and honoured him, and so do we.
Then his Lord chose him and turned to him in forgiveness and guided [him]. (20:122)
Indeed, Allah chose Adam and Noah and the family of Abraham and the family of 'Imran over the worlds - (3:33)
In the Qur'an, Allah addresses all of humanity by virtue of our being the Children of Adam:
O children of Adam, if there come to you messengers from among you relating to you My verses, then whoever fears Allah and reforms - there will be no fear concerning them, nor will they grieve. (7:35)
And We have certainly honored the children of Adam and carried them on the land and sea and provided for them of the good things and preferred them over much of what We have created, with [definite] preference. (17:70)
Those were the ones upon whom Allah bestowed favor from among the prophets of the descendants of Adam and of those We carried [in the ship] with Noah, and of the descendants of Abraham and Israel, and of those whom We guided and chose. When the verses of the Most Merciful were recited to them, they fell in prostration and weeping. (19:58)
I am not sure what you mean by the whole Adam and Eve story, but we believe God created them both, and that they lived in the garden, and that Allah taught Adam the names of all things. Allah ordered the angels and jinns to bow down to him, but Satan refused, as he thought he was better than Adam, and promised to try to mislead Adam and his children. Allah told Adam and Eve not to eat from a certain tree. They were warned that if they ate from the tree, they would be wrong-doers and that if they listened to Satan, he would cause them to be deprived of the garden. Satan had said that he would do his best to mislead humans. Placing Adam and Eve in the garden gave them their first experience of dealing with this challenge. Satan tempted them both, and they both ate from the tree, and immediately regretted what they had done. They both repented and prayed for forgiveness (and the prayer they used is one still used by Muslims today in their supplications), and both were forgiven by their Loving, Merciful Lord, nothing further was needed, and Allah honoured and guided him and they left the garden to live on earth.
This lesson for all the descendants of Adam tells us that we, as humans, have the freedom to choose, to err, and to repent sincerely if we wish, and should we do so, we will find Allah, Forgiving, Merciful. And that we need to be careful of Satan trying to mislead us. Indeed on the day of judgement he'll say to those who followed him or those he lead astray:
And Satan will say when the matter has been concluded, "Indeed, Allah had promised you the promise of truth. And I promised you, but my promises were false. I had no authority over you except that I invited you, and you responded to me. So do not blame me; but blame yourselves. I cannot help you, nor can you help me.. (14:22, part)
Yes the Qur'an states that Allah created the heavens and earth in six days (but Allah knows best if they are 24 hour days or other periods of time). Why does that not seem true to you? Allah did not however, rest on the seventh day, far removed be He from that.