Did Carol Kill Lizzie?

Lizzie tells Tyreese and Carol that she has spared Mika's brain so that she will come back as a walker, and asks them – at gunpoint – to leave Mika's body alone. Carol later kills Lizzie after taking her outside to "look at the flowers" to distract and calm her.

Is Lizzie a psychopath walking dead?

If you look closer at what we know about Lizzie and the science of psychopathy, it becomes clear that Lizzie was struggling with the zombie apocalypse but she wasn't a psychopath.

Did Tyreese know that Carol killed Karen?

That's what was happening to Carol — her mindset — she was so hell-bent on protecting these children that she lost a bit of something, and that was her nurturing aspect. She was missing a lot of stuff because her eyes were so set on survival. Carol finally told Tyreese that she killed Karen.