1. Yes the cook thing's that one instance of sexism we have mentioned already before, the other's no different than other people's remarks, Naru's gender's part of the existence Naru experiences but the idea it is the focal point's just incorrect.
Cooking's also not something attributed purely because of Naru's gender, as just an assumption, but something we actually see happen irrespective of that context.
Just like the medicine, or tracking, or fighting.
2. Not the whole party's refering to Taabe not Naru obviously, she's smug when saying that but then loses that in shock when she sees they have him captured.
3. It is because the leader of the fur trappers simply underestimated Naru, just like when he saw her as an animal to be prodded when in the cage, and underestimated the ferocity of a Comanche Warrior even then. I think that has honestly more to do with underestimating native peoples than gender.