From my experience, if you sound sincere about your question, and it's written in a way where you don't sound as if you are an apostate, they will reply to your letter and send you a personalized letter back with the best answer they can give to your question. It may not be the answer you want, just the answer they can give. They have standardized responses for most inquiries, but they will also try and address any specifics your initial letter may have asked. If they can't answer something specific about your letter, they will just ignore the question or specific issue raised.
However, as others also stated, if you include your personal information, such as your name and congregation, they will also send a copy of their reply to your BOE. This isn't always done to "rat you out" with malicious intent, necessarily. The main motivation is so that they are aware of whatever inquiries you might have, since they are supposed to be the first people you direct questions to, since the branch does get quite a lot inquiries every year from individuals. So basically, they rather you ask the elders your questions first so they don't get swamped with unnecessary inquiries from individuals. Plus, they really do think elders should take care of their own congregation members, and will therefore keep them in the loop anytime an individual sends in a question that they feel should have been directed to the elders first. In addition, if you do ask a question that the elders can't answer, they prefer that the elders send in the inquiry to help cut down on individuals sending in personal inquiries.
As to whether or not the elders would address their response with you, it would probably depend on the nature of the inquiry, and whether or not they feel they should get involved. If the inquiry was about something that was not important, I doubt the elders would even address the response with you.