I disagree with the answers stating that "cool" and "stupid" are used as adjectives here. If someone "plays it cool," that person might be a "cool" person, but that is not what the sentence is saying. And "cool" can't be a predicate adjective because "play" is not a linking verb!
Moreover, the first comment on the question, stating that the sentences don't change much when the adverb forms of "cool" and "stupid" are used, seems to me to be exactly correct:
She plays it coolly.
He acts stupidly.
Therefore, I believe this is simply an instance in which adjectival forms are used instead of adverbial forms, probably because they simply "sound better".
Another common example of this is the use of "bad" instead of "badly." Rarely does anyone one say "I feel badly" to describe how they feel (though I've seen a grammar book recommend this!), but it seems unlikely that in the sentence "I feel bad", the speaker is trying to describe themself ("I") with the descriptor "bad".
As for "proudly", that's just a standard adverb modifying a verb, which is perhaps the primary function of adverbs.