Looking at Haruki post, she didn't really say the fan was her actual gem weapon so it could be a human fan that just seemed right to have. A sixth sense kind of thing.
I also did leave an option open that they could be awake for awhile before venturing so she might have found small information in her time alone to know at least what gems are...
Also, I don't think Carnelian knew of other gems but could be hopeful she isn't alone though you are right to question all that you did since I am merely speculating