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Iago first mentions his suspicion that Othello has had sex with Emilia in Act 1 Scene 3, ln 368 (in the Cambridge School Othello--might be different line numbers in your text), when he says, "I hate the Moor, and 'tis thought abroad that 'twixt my sheets he has done my office." He refers to his suspicions again in Act 2 Scene 1 line 275, when he says, "For that I do suspect that the lusty Moor hath leaped into my seat". Both of these are vague references, and easy to miss. You have to know that both doing someone's office and leaping into someone's seat are euphemisims for sexual activity. Actually, almost everything in this play is a euphemism for sexual activity, so it's not much of a leap. The most important thing is that it is NOT evident, at least not for anyone but Iago. He says himself that he has no proof, but because he has heard the rumour he will assume it is true. (Emilia, later in the play, chides him for suspecting that she had an affair with Othello). These are the only three pieces of evidence that I know of that link Othello to Emilia. The other person that Iago suspects of being with Emilia is Cassio--he mentions this in the same soliloquy in Act 2 Scene 1--he says, "for I fear Cassio with my night-cap too". Again, the language is vague and easy to miss. A "night-cap" is a hat one wears to bed, or a drink that helps one sleep; however, in this instance, it is an action that puts one (men anyways) to sleep.
In act 3 sc 4 Emilia Desdemona asks Emilia (point blank) where the handkerchief is. Emilia says she does not know. We can see here Emilia exhibiting a similar duplicity as her husband. I'm not exactly sure what line you are referring to in the second part of your question. Desdemona says she never gave Othello cause to doubt her. Emilia answers,
"But jealous souls will not be answer'd so;
They are not ever jealous for the cause,
But jealous for they are jealous. 'Tis a monster(170)
Begot upon itself, born on itself."
Jealousy is, she says, is simply entrenched within the character of a person, it has no cause or reason to it.