The difference between serious and seriously is that the former is an adjective, while the latter is an adverb. We use adverbs to modify verbs (hence Play seriously = [Verb] [adverb]), but adjectives to modify nouns (hence Serious play = [Adjective] [noun]). Since fast is both an adjective and an adverb, run fast and fast run are both correct.
It would be erroneous to suppose that adjectives can be applied to verbs. In the first of your examples, though fast is correct as an adjective, it is also correct as an adverb. The second example is fallacious since clean is not an adverb, making play clean incorrect.
As to the list, what you would end up finding would be adjectives which are also adverbs, such as fast, since that type of word is the only type which can modify both nouns and verbs. Clean, as before stated, would not be on the list.
Also, the tag is correct. Adjectives modify nouns, and adverbs modify verbs.