And it wasn't Carol's first instinct to kill Lizzie, she first wanted her and Lizzie to go off on their own while Tyreese and Judith continued to Terminus. But they didn't seem to think that was an ideal plan either.
The deal with Lizzie was that it was an imperfect, no-win, scenario. And it was agonizing for Carol to do what she did. Maybe that's part of what steeled her and made her go down this particular path, but it really was a case of what else could they do? Lizzie obviously had issues, issues that went beyond her simply not "getting" that the walkers weren't still people. She needed to be medicated and have counseling. Neither of which are in abundant supply right now. Lizzie was a problem and still a threat, what else could they do? It's hard to say they made the "right" choice but it really was a case of there not being much else at their disposal. Lizzie could not be trusted around other people. Even herself or just Carol. Leaving her alone is just as good as killing her, so...
Carl was a whole other case. The kid he killed in the woods outside the prison Carl saw as a threat and was able to reasonably justify it. I sort-of agree with him and think Rick and Hershel were out of line in thinking what Carl did was wrong. Carl asked the kid to drop the gun and the kid was trying to hand it over. This means for Carl to retrieve it he'd have to drop his own guard to get it giving the kid an opportunity to reclaim the balance of power and become a threat. When Carl lowered his gun the kid could step-in and overpower Carl now putting Carl, Herschel, Beth and Judith all at risk or when Carl lowered his gun the kid could re-fortify himself with his own gun and take Carl out, putting Herschel, Beth and Judith at risk.
Carl didn't try and to "justify" or rationalize what he did, he knew what he did wasn't "good" but he felt it was the right choice at the time. He'd seen others, including himself, not seize an opportunity to take someone out and for that to backfire. Most recently and more notably Rick not taking out one of the prisoners who later was able to set into motion the chaos at the prison that led to Lori's death.
But, Carl wasn't a lost-cause. He didn't think he had to kill people just to kill people, just kill people who could be threats, even if they probably weren't. A lot of air in there to say whether or not that's the right way to do things in this world but certainly not a case where Carl is enough of a threat that he needed to be stopped at any cost.
Lizzie was a lost-cause. She felt that the walkers were still "people" or at the very least not something to be afraid of and kill. (She really seemed to think of them more as pets. Like dogs.) She killed her sister and was moments away from killing a baby thinking this, this was after numerous cases of her being told to be afraid of the walkers and after numerous dangerous encounters with them. There wasn't much else to do as she clearly had mental issues that could not be dealt with in this world.
Lizzie and Carl? Very, very, different scenarios. Carl was recoverable and still had some emotion and humanity in him. And he still does. Lizzie was gone. She gleefully killed her sister and was ready to kill a baby that couldn't even walk yet because she felt that walkers were still something of sentient beings worthy of keeping around. There was almost nothing that could have been done with her.
"The Grove" is one of my more favorite episodes of the series because it's just such an F'ed up situation there's no good answer to and, ultimately, I think Carol made the "right" (or slightly less wrong for want of a better term) choice. At you can tell it broke and crushed her. And Tyreese taking her confession and forgiving her for killing Karen was a great scene too, as he likely saw Karen as something similar. She needed to be killed for something of a greater good. (Though I think Carol was wrong there due to a painfully huge misunderstanding of how viruses work.) And the no-win scenario with Lizzie is just really, really fantastic stuff.
But I think it's another bath on an arc to make Carol end up self-destructive as she becomes a darker and darker character, a much steeper arc than the one Rick is on. I'd almost argue Rick's arc isn't even leading him into being a dark character but just a "do what's necessary" leader. As we've seen him still have a degree of humanity and reason and logic in him.
Rick's first instincts going into this place was mostly one of caution and mistrust, he's slowly starting to maybe see the place as the haven it is and the people inside not as threats but as a community sheltered from the apocalypse.
Carol's first instincts were to not trust anyone and to start planning a subterfuge in order to infiltrate the community and potentially strike out if needed. That's almost psychotic of her or at the very least very devious and cunning. It's almost what The Governor was doing. The Governor putting on this good-ol' boy, aw-shucks charm of being an unwilling leader of a community when really behind it all he was much more devious and ruthless.
Would Carol really carry out her threat on the little boy? Knowing the places she's been as a person over the course of these events that's hard to say. I don't think it's clear if it was just talk to make the boy piss his pants and not tell anyone or if it was a genuine threat. And not knowing is what puts Carol on that darker arc because where she's been makes it seem probable she could carry this action out.
Rick's means of threatening someone, Aaron, is to tell him he has an hour for his story to work out. And Rick admits to being a "good guy" but still willing to carry out killing someone who may be a threat. Now, we know Rick enough that he likely would have killed the guy if no one came back within that time, meaning the guy was likely lying. But his killing of Aaron in that case would have been no different than when he killed the guys in the bar in Season 2 ("Nebraska") or Tomas in Season 3 (one of the prisoners who tried to kill Rick via walker.)
Carol's means of threatening someone, a child, is to scare the child to shit and suggest she's going to kill him in the cruelest possible way in this world.
Carol's on a dark arc and it'll be interesting to see how it pans out for her in the long run. I, honestly, don't think it's going to end well for her because she may be at an irrecoverable point.
