Learn English – word for “conjugating” an adjective

As far as I know, irregularity correlates to frequency of use: as a paradigm ( / set of related words) is more commonly used, it will be less regular on average. Suppletion is an (extreme) type of irregularity, and so it is more common with frequently used words.

The most common adjectives and adverbs in all languages I know have more suppletion than their less common counterparts (good v. well). Pronouns and common verbs are very frequent in many languages, and they are also often suppletive to various degrees, or at least very irregular (cf. I v. me and am v. be).

The reason is probably that regular paradigms are easier to remember. If you have a certain word with old forms and new forms, and this word in not very frequent, you are more likely to forget some of the forms, and iron out some irregular features. But if you use the irregular forms every day, you are more likely to "remember" them, they will be more strongly rooted in the linguistic network of your brain, whatever that may be.

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.