Translation:How many people are in your family?
Point of confusion: I read kazoku is for when talking about my family, but the more polite gokazoku is when talking about someone else's family. In DL, I see the polite nouns being used to refer to one's own family members, and that's fine, but in this example, the casual term is being used for someone else's family member. How firm is this convention, and will DL correct me if I start doing this? Does anyone know?
何人（なんにん） does not mean "what people." It means "how many people." 人 is a counter just like 本, 冊, 枚 and we ask quantities with 何本, 何冊, 何枚.
おいくつ means "how old are you," and it does not mean "how many." いくつ without お means "how many" but it can never be used to count the number of people.