My goal with the first line was to make it clear that Leia is fully aware of everything that Rey is experiencing, desperately trying to get her to admit the truth so that she can teach her a lesson. But instead Rey chooses to lie. It’s also another scene that is extremely awkward and feels more like a one-sided conversation in the original. Even when Leia finally says something to Rey (who clearly needs more emotional support than that), it’s evident that it’s taken from her asking for a status report on Poe and BB-8 from TFA. The goal here is to recontextualize the line so that it actually feels like a natural part of the conversation. In this case, you could place both sentences together and they make sense independently of Rey speaking: “What’s troubling you, Rey? Don’t tell me what things look like, tell me what they are.”

Agreed on the second line’s delivery, I’m going to continue adjusting that for a take I’m more satisfied with.

As for your third point, that’s certainly part of it. Basically, she chooses the path that she feels is least likely to lead her to the dark side and hatred (being a public figure like her mother). Instead, Leia still lost everything and everyone. Prime dark side stuff. But, regardless, she choses love in the end over hatred for all that happening. So it’s up to Rey to deny that being a Jedi and taking action can be something which makes you fall (why she’s on Ahch-To in the first place), and instead choosing love/acceptance is a universal trait anybody can do to overcome the darkness they struggle with.

Rey takes this lesson to heart when she says this to Palpatine later on: “All you want is for me to hate. But I won’t. Not even you.”

EDIT: How about this for the second line - “Leia sensed his nature in you…” I think this would evoke the nature vs. nuture debate quite well. What is better - to be born good, or to overcome your evil nature through great effort?