"I do not know the woman."
Translation:Ik ken de vrouw niet.
Well, I tried to figure it out myself and found a rule in a book (the book is in Russian so this is my translation) that says that "geen" is used to deny a noun and it replaces indefinite article "een". In that sentence the woman has definite article so that might be the reason. But may be not :) I hope someone with better knowledge of Dutch will come to the thread and answer.
It does matter. In not like "you" that if you don't know it's plural or singular it should be accepted. In this case there is no reason to use mevrouw so you shouldn't use it.
You use mevrouw not as a synonym for woman. Mevrouw is used for a direct adress mainly.
Very young kids might say I zag een mevrouw (like up to age 6 perhaps or atleast 4)
As an adult you might refer to someone as a mevrouw but only in very specific situations. Usually when the woman is old and posh.
Otherwise when talking about a female you just say vrouw.
As a new learner you basicly only need to know
Mevrouw=Miss as in adressing a stranger formally. Miss you left your wallet, miss can I ask you something