"Does Esperanto have its own culture?"
Translation:Ĉu Esperanto havas propran kulturon?
I would be inclined to say
- ...havas sian propran kulturon? = ...have its own culture?
- ...havas propran kulturon? = ...have a culture of its own?
That is, the first sentence pair implies that one single culture of its own exists and asks whether Esperanto has it.
The second one does not imply that this culture exists, and so I think it's a better way to ask about it.
For example, I would ask "Do you have brothers or sisters?" rather than "Do you have your brothers and sisters?", or "Does Mary have brown hair?" rather than "Does Mary have her brown hair?".
That said, in English "its own culture" seems reasonable to me even without this necessary implication -- perhaps because of the way "own" works. ("Do you have your own room?" and "Do you have a room of your own?" both sound fine to me while "Do you have your room?" does not.)
I don't know how the majority of fluent Esperanto speakers treat propran but I think the version without sian sounds better if you are asking about the existence rather than presupposing that it already exists.