At the heart of the points of contact between The Thousand and One Nights and the Qur'an stands a particular worldview that informs both texts. Let's look at this worldview to help you get started on these questions.
The Islamic worldview expressed in the Qur'an centers around God as the Creator and Ruler of the universe. God is omnipotent and in control of history, and people are subject to a particular destiny according to God's will. Yet they are also morally responsible for doing good and rejecting evil in their lives. They are also responsible for caring for each other and being faithful to each other and to God. There is, therefore, a balance between destiny and free will in this worldview and between God's will and the human response to it.
This worldview is also prominent in The Thousand and One Nights. We can see it in the frame story of Shahrayar and Scheherazade. Scheherazade is determined to stop the sultan from killing his wives, so she comes up with a plan to tell a thousand and one stories, stopping in the middle of each one so that the sultan is so curious about the ending that he does not kill her in the morning. Scheherazade is not just rolling over and submitting to destiny. She is participating in shaping her destiny and that of other women through her free actions.
These tales also strongly reflect on the themes of good versus evil and of faithfulness. We can see the former in stories like “Aladdin's Lamp,” where good triumphs over evil. Faithfulness is at the heart of “The Three Princes and the Princess Nouronnihar” in which the three princes decide that it is much better to faithfully save Nouronnihar's life then to compete to win her hand.
Finally, the stories show the theme of fate and destiny and the human response to fate and destiny in tales like that of Sinbad the sailor who seems to challenge destiny continually and yet manages to survive through his cleverness.