"What is your story?"
Translation:आपकी कहानी क्या है?
no, kya comes directly before the verb when asking a 'what' question. Subject + object + kya + verb aapki kahani kya hai? literal translation: your story what is = what is your story?
कहानी is indeed feminine and आपकी takes the gender of its object (कहानी) so you would say आपकी कहानी क्या है? even if you were talking to a man.
It's है because कहानी is singular.
In case of आपकी कहानियाँ क्या हैं? ('What are your stories?') there is a हैं.
आप is the honor way to address a person, I learned that I should use the plural form.
That's correct but the subject of this sentence is not 'आप'/'you'. It's 'आपकी काहानी'/'your story'. So, the verb will conjugate according to काहानी.
This is similar to how you should say 'Their house is nice' with a 'is' even though 'they' usually goes with the plural form 'are'.