"Do you have a family?"
Actually, if you wrote かぞくがいますか？you would make the family, rather than any implicit "you", the subject of the sentence. That is what が does, mark the (non-topic) subject. So かぞくがいますか would mean "Is there a family?", while the topicalized かぞくはいますか would be more akin to "Does the family exist?", with an implied reference to the listener turning "the family" into "your family" and thus "Do you have a family?". (Or at least I think so -- there could well be deeper levels to this.)
I agree that the sentence, as is, would mean "Is there a family" since は implies that family is the subject rather than the object. However, it could simply have idiomatic meaning in that "Is there a family?" in Japanese actually means "Do you have any family members?" Idiomaticity is a rather common phenomenon in the Japanese language, so I would not be surprised if this were the case here.
I think it's because in this context, the questioner asks the questionee, "kazoku wa imasu ka?" So it'd be the same as "hey, do you have a family?" That's how it intuitively came to me.
If it's "kazoku ga imasu ka?" The questionee addresses a wider audience or no audience and then asks "hey, does a family exist?"
Whether it's a question or a statement makes no difference with が and は. が is used to draw attention to the subject and can only be used on the subject, you use it when the information is new. は can actually replace many particles, and it's kind of like using a pronoun instead of saying the actual object. Duolingo's strong point isn't grammar, so I highly recommend you go elsewhere for that.