He goes into their room while she is sleeping and then she wakes up and after she does he smothers her.
Iago had lied to Othello and convinced him that Desdemona was secretly having sex with Cassio.
No. Othello did not kill Iago. He killed Desdemona (he thought he did justice). Then, the truth came out. Othello charged at Iago but was stopped by others at that scene. Then Iago ran away, and Othello killed himself; died upon a kiss on Desdemona.
Othello was a frequent guest at the house of Desdemona's father.
Brabantio is Desdemona's father. He becomes very upset when Desdemona elopes with Othello and accuses Othello of committing witchcraft in order to woo his daughter. Othello denies any wrongdoing and Brabantio warns Othello that if Desdemona deceived her father, she can deceive Othello as well. Brabantio dies from a broken-heart resulting from Desdemona's marriage.
No, almost certainly not. Desdemona was a loving and true wife and would not have given Othello any reason to kill her. And Othello was basically a fair-minded man as the play shows before Iago began to spin his web of deceit. Not only that, but Othello truly loved Desdemona and under normal circumstances would not hurt her. Everyone is shocked when he strikes her in public not only because it is shocking behaviour but also because it is uncharacteristic. Without Iago it is as likely that Othello would kill Desdemona as it is that she would kill him.