So a few things here!
The NHIE choices do affect stats, just not the ones you might expect to see shift. Picking “I cheated” doesn’t change your relationship with Mattie because you didn’t cheat on Mattie, you’re essentially strangers who have known each other for a few hours at this point. What it does affect is her trust in you (she lost 4 trust points, which feeds into the villa trust calculation), her pref (down 4, so she’s marginally less inclined toward you over the other islanders), and your popularity with the public. Your relationship with her, the romantic and platonic tracks, reflects what you’ve actually done/said to her, and you haven’t done anything to her yet. She just learned something about who you are. That’s different.
Also, if you’re playing a 28 year old who cheated once at 18, that’s a decade of growth. People change. The game accounts for that, and later, you’ll get to talk about your reasons for cheating, the amount you cheated, etc, which is more important than just cheating.
As for drinking when you say you’ve never done something, that’s not how Never Have I Ever works 😂 If you say you’ve never done it, you haven’t done it, so you don’t drink. That’s the whole game. The option to drink when you pick the prompt isn’t there because you’re the one asking the question, not answering it. If you want the thing to apply to you, don’t pick the prompt and then it can apply to you later.
And yes, the public relations tab moving during the drinking game is correct and intended! Every round is being watched. You’re always on camera.