What is the purpose of "czy" here? Would "jesteś kobietą?" not be enough?
From my limited experience, "czy" indicates a yes/no question.
I think "czy" could be dropped, if you intone the final syllable of the final word in question form (i.e. rising tone).
That's exactly the case. Czy indeed indicates a yes/no question. It can also initiate a subordinate clause: I ask myself whether you are a woman = Pytam się czy jesteś kobietą. Note that when it initiates a subordinate clause czy is necessary.
You're right, but czy can also give emphasis to the question.
So jesteś kobietą is also gramatically correct?
Yes, it is. Also note that the use of czy to start a yes/no question is somewhat formal and most Poles don't use in everyday speech, but it's nothing unusual to use it.
Is czy similar to "est-ce que" in french?
Yes, it is.
this is like russian ли, right?
A bit, only that as far as I understand it is not used very often in Russian anymore (at least it wasn't used often in the course), and in Polish it's very common.