Agreed. In BrEng I think this is the only correct answer - fast is an adjective and quickly an adverb. I think fast is acceptable as an adverb in AmEng though, and probably casually fine in BrEng too.
According to the Oxford dictionary, 'fast' is an adverb as well as an adjective in British English.
They dance very fast sounds weird to me. "Quickly" at least should be accepted.
Is there any difference between adverbs and adjectives in Hindi? Could तेज़ be used as either?