"What is your story?"
Translation:आपकी कहानी क्या है?
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'.
No. You use अपनी when the sentence has an already established subject and you are referring back to it.
For example, in a sentence like 'आप अपनी कहानी बता रहे हैं' (You are telling your story), the subject is आप (you). Since we want the pronoun to also refer to the same entity ('your'), instead of repeating the subject by using आपकी, we use अपनी.
The subject of the sentence 'आपकी कहानी क्या है' is 'आपकी कहानी' so there is no pre- established subject to refer back to. That is why अपनी cannot be used.
Note that all imperative sentences (requests and commands) have one of the forms of 'you' as an implicit subject. So, you can say 'अपनी कहानी बताइए।' (Tell your story). Since this is an imperative, आप is the subject of this sentence even if it's not explicitly written. This is the reason you can use अपनी to refer back to it.
Can this be used idiomatically the same way that it is an English? In English this could mean a literal story that someone wrote, or a "story" of their experiences, or even an explanation/excuse for something.
I swear I've heard कहानी in a Bollywood song before, but I can't remember which or what the context was.