gramatická analýza: Vykuliv nadšeně oči a zpěvavě protahuje dlouhé slabiky recitoval.

The main conjugated verb is recitoval. The subject for the verb is therefore masc. sg. Vykuliv is a past verbal adverb, masculine sg form of vykulit. Protahuje is a present verbal adverb, masculine sg form of protahovat. Since they are both in the masc. sg., we can conclude that they both add information to the main verb recitoval, explaining how the person was reciting the poem.

Try paraphrasing the sentence using independent clauses: "he rolled out his eyes and then recited, and at the same time he stretched the long vowels in a singing fashion"

For "vykuliv oči" it is also possible to use a prepositional phrase "he recited with rolled-up eyes."