Give examples in Othello about how friendship is shown, the penalty for betrayal, who remains loyal, and what the play says about friendship &...

There is very little true friendship in the play.  None of the men are friends.  All are too paranoid about their women and too worried about their reputations to enter into an honest friendship.  Cassio might have been a friend to Othello, but Othello is so insecure around the younger, white Cassio; Othello's uber-paranoid that Cassio will steal his white trophy wife (even before Iago brings it up).

Emilia and Desdemona have the only honest, friendly conversations in the play.  Yet, they have differing opinions of a wife's and husband's duties in marriage.  Desdemona says she could never betray her husband, not for the world.  Emilia says that men are stomachs and women their food.  They are friendly, but not friends.  The social hierarchy does not permit friendship: Emilia is her servant, not her friend.  If Emilia was a better friend, she might have stopped the honor killing. Only after Desdemona has been killed and she stabbed does Emilia realize how bad a friend she was.

Iago is the first to feel betrayed by Othello.  His reputation feels insulted because a) Othello slept with his wife and b) he chose Cassio as his lieutenant.  I don't think Emilia betrayed Iago and slept with Othello; Iago, a racist in denial, fears Othello's hypersexuality and so he invents the first reason to motivate himself.

Brabantio feels betrayed by his daughter, who secretly married Othello.

Roderigo feels betrayed by Iago, who promises Desdemona as Roderigo's prize.  Roderigo is betrayed on an emotional level because Desdemona rejected his suit and on a monetary level because Iago steals his life's savings.

Othello feels betrayed by nearly everyone except, ironically, Iago.  He feels Cassio betrays him by getting drunk and disturbing his honeymoon night.  He feels Desdemona betrays him by losing the handkerchief.  He feels Emilia betrays him by covering up for Desdemona and Cassio's supposed affair.  Only after he kills his wife, does Othello know that Iago had betrayed him all along.

Emilia feels betrayed by Iago, at the end, because she feels complicit in Desdemona's death.  Having been shunned by her husband for so long, Emilia obtained the handkerchief to get in Iago's good graces.  Her strategy backfires.  Iago kills her.

Iago feels betrayed by his wife at the end.  He expects her to be the quiet, submissive wife.  Instead, she speaks out against him.  And he kills her for it.

Some quotes.  Othello says, "Yet she must die, else she'll betray more men."

Desdemona says, speaking of Cassio, "Alas! he is betray'd and I undone."  This further motivates Othello to kill Desdemona, as her betrayal seems complicit with Cassio's.

Instances in which words of friendship and fidelity are being masked by acts of betrayal: everything that Iago says!  Namely, to Othello "I am your own forever."