@kawaii123456 well, the action by itself it's surely peculiar but what can they think about her with just the little information they have? Their only answer would be what they already know: that she's smart. Who knows how much and who knows what she wants, that's still a mystery, but her actions and how close she's staying to the Crown Princess' side, it indeed make it seems that she's loyal to her; so if other people want to doubt her, they'll have to wait more time to discover what's going on, and that's An Qi's advantage.
In general, I don't think it's a big deal because in the palace you must be aware of everything and everyone anyways.
About the poison stuff, I also wonder how it works because it hasn't been stated clearly in any of the chapters so far, but I've seen similar cases in movies so my guess is that maybe her body's not the only one who can cure him or works for him, but she has something special that makes her survive (not like other women) and that's why he's still using her (aside from the other reasons that we already know).
So far I can't really tell if what the demon Prince says is true or not, but I guess An Qi doesn't have many options at the moment; for me, it seems like she really doesn't care much about if she's been poisoned or not or if she's the cure for him, just as u said, she's focusing her attention in some side-quests (which makes me wonder some things as well about her motivation, lol). She just knows that he's powerful enough to kill her and that he's moving the strings to kill everyone else in the Palace, so she's probably just taking her time to figure it out some things. Idk but I wouldn't dare to move my cards against him that easily knowing that neither. Ofc, unless I've found someone certainly stronger and smarter than him or a way to escape from there.
Edit: Tnks for the chapters guys. :b