Yes, and the sentence is exactly as written. I'm just supposed to chose. That is why it is confusing.

Answer 1

Note the spelling of "grammar"!!

You wrote:

Money might be sewn into a woman's or women's coat linings.
If the word "a" needs to be in this sentence, then you need to choose "woman's" since it's singular. The pairing "a women's" won't make any sense at all!

Worry haunted the refugee children's or childrens' eyes
Since there is no such word as "childrens'" the obvious choice is "children's." Do you know why it's obvious?