Emilia. It was proper to gain access to a lady's presence through her maid. This also enabled Emilia later in the play to say that Cassio and Desdemona were never left alone together, and so could not have had an affair.
He tells Cassio to ask Desdemona for her help to get him reinstated.
Emilia overhears Othello and Desdemona discussing Cassio. Desdemona is pleading on Cassio's behalf to Othello, hoping to persuade him to reinstate Cassio as his lieutenant. In the conversation, Desdemona is trying to help Cassio regain his position in Othello's esteem.
If Desdemona asked a favour on Cassio's behalf, it would look to Othello like she was doing it out of an improper love for Cassio. The harder she argues, the more suspicious it seems to Othello.
Iago tells Cassio that he must persuade Othello to reinstate him. He says "the general's wife is now our general" and so the best way to persuade Othello is to get Desdemona to plead on Cassio's behalf. Since Iago has also told Othello that Desdemona and Cassio are having an affair, the harder Desdemona pleads, the worse it is going to look for both her and Cassio.
Cassio sees Desdemona as a sort of secular Virgin Mary, as from the quote, "O behold...Hail to thee lady!". On the other hand, Iago sees Desdemona as competition for Othello's affections.
Iago says that he was sleeping next to Cassio and that Cassio was having a dream about making love to Desdemona.
Iago advises Cassio to ask Desdemona to plead on his behalf. This will help Iago poison Othello's mind, as Desdemona will appear to be doing so out of her (illicit) love for Cassio.
Cassio kisses Desdemona's hand. Iago says, " He takes her by the palm: ay, well said, whisper. With as little a web as this will I ensnare as great a fly as Cassio."
Because Othello is unsure of Desdemona's actions. He is afraid of losing Desdemona to Cassio.
When Iago tells Othello Cassio slept with her.
Iago realizes that Cassio greets Desdemona romantically, and with a kiss on the cheek. Supposably that's how you greet the women politely.
Iago sets out to trap Othello by several means. He persuades Desdemona to plead for Cassio and persuades Othello that Desdemona pleading for Cassio proves she is in love with him. He steals the handkerchief, persuades Othello that she has given it away to Cassio, then gives it to Cassio himself. He gets lucky as Cassio gives it to Bianca and she returns it in Othello's sight. He talks to Cassio about Bianca, telling Othello who can see (but not hear) that he is talking about Desdemona. All of this circumstantial evidence bolsters the outright lies that Iago tells about Cassio admitting to having an affair with Desdemona.