When they hear that the Duchess has given birth, Ferdinand reacts by totally flipping out. The Cardinal, on the other hand, plays it cool, and tells Ferdinand that if he wants to get revenge he's got to chill out. The Cardinal frames anger as dehumanizing, and this comment foreshadows how Ferdinand, who's all about the "deformed, beastly" anger, ends up transforming into a werewolf.
Excellent, as I would wish, she's plagued in art. (3.1.107)
Why, once he gets ahold of her, does Ferdinand resort to psychological, rather than physical torture? He oh so obviously intends to kill her, and at this point he could be putting the screws to her in a very literal way. But he prefers to "plague her in art" as a way to "bring [her] / by degrees to mortification" (4.2.166-67)—why?
Bosola: What do you intend to do?
Ferdinand: Can you guess?
Ferdinand: Do not ask, then.
He that can compass me and know my drifts
May say that he hath put a girdle 'bout the world
And sounded all her quicksands. (3.2.83-86)
Ol' Ferdo sure thinks an awful lot of himself and his crafty skillz. Bosola, sassy spy that he is, immediately tells Ferdinand he's not as tricky as he thinks he is. This is a neat line to pair with the Cardinal's claim in Quote #7 that people who know a prince's secrets (the "drifts" Ferdinand refers to) "need have their breasts hooped with adamant / To contain them" (5.2.253-54).
What rests, but I reveal
All to my lord? Oh, this base quality
Of intelligencer! (3.2.321-23)
The Duchess has just let slip to Bosola that Antonio's her husband. He knows he's gotta go tell Ferdinand, but he doesn't do so right away, waiting for a moment and wondering what's holding him back. Bosola's just discovered that the Duchess has married (and thereby "rewarded") Antonio for no other reason than that Antonio's a good, worthy man, and thereby realizing the dynamic that Bosola himself both most derides and fantasizes about: the dream of reward for virtue and service. He doesn't want to bring down the Duchess, but his compulsion to fulfill his duty to Ferdinand overwhelms his moral qualms.
Where I am a man
I'd beat that counterfeit face into thy other. (3.5.119)
The Duchess realizes that she's been betrayed by Bosola, who's about to take her to Ferdinand. Bosola frequently draws a distinction between his real self and his intelligencer self, and the here Duchess wants to smash the two together. This two-facedness—between the real and the "counterfeit" selves—is part of the "base quality of the intelligencer" mentioned in Quote #3.
Bosola: Must I see her again?
Ferdinand: You must.
Bosola: Never in mine own shape,
That's forfeited by my intelligence
And this last cruel lie. When you send me next
The business shall be comfort. (4.1.128-33)
Bosola's reached the end of his (considerable) tolerance for evil-doing. He's gotten to the point where he realizes he can't neatly separate his work for Ferdinand from his own set of values and understanding of right and wrong. He can say that he's just doing his job, just taking orders but he knows that his "own shape" really has been corrupted by his work as an intelligencer.
[…] I'll go in mine own shape […] (5.1.68)
Despite everything that's happened, Antonio's convinced that if he can just explain the truth to the Cardinal, the Cardinal will see reason, they can hug it out, and everything will be copacetic. Antonio's described as relentlessly honest, a quality which makes him a good guy, but unsuited to the world of deceit and intrigue he enters as the Duchess's husband. Contrast his intent here to go "in his own shape" to the Cardinal with Bosola's many different "shapes."
[…] think what danger 'tis
To receive a prince's secrets: they that do
Had need have their breasts hooped with adamant
To contain them. (5.2.251-54)
Julia's badgering the Cardinal to tell her what's on his mind (um, the execution of the Duchess and her kids?), and he keeps trying to tell her, "girl, really, you don't want to know; trust me, you can't handle it." One of the neat things about this line is the way in which the Cardinal is characterizing "a prince's secrets." In trying to find out what he's thinking, Julia isn't just trying to be close to him (in the way a woman could be close to her lover), she's trying to become his princely peer.
Cardinal: […] Think you, your
Bosom will be a grave dark and obscure enough for such a secret?
Julia: It lies not in me to conceal it. (5.2.263-66)
Julia thinks she's ready for this (evil, manipulative) jelly, but soon finds out that she really, really isn't. At the behest of Bosola, she's trying to find out what the Cardinal's worrying about, promising him that she's totally capable of keeping his secrets. Julia's attempting to become the Cardinal's confidant in addition to being his lover, but ultimately she's not cut out to run with the big dogs, and it costs her life.