"I like the belly dancing."
Translation:J'aime bien la danse du ventre.
It would depend on the context.
With "the", the speaker would be referring to the specific belly dancing currently in progress or otherwise implied by the context.
Without "the", the speaker would be referring to belly dancing in general.
"danse du ventre" is simply how the French say it as standard. "La danse de ventre" would probably be closer to "the belly dance" (versus "belly dancing" in general).
Why is J'aime la danse du ventre marked wrong? Why is the use of "bien" mandatory here?
"Bien" isn't mandatory here, though it may make the sentence sound better/smoother to a native speaker.
Something to report, I think.