Poēta puellae fābulam narrat.
Does every word in here have a case (i.e. dative, accusative, nominative, etc)?
Every noun in your sentence has a case, but the verb does not. This is always the situation.
Here poeta, puellae and fabulam are nouns and have cases (nominative, dative, accusative). The verb narrat is also in a specific form, but not in any case. The verb is in third person singular, more specifically that of active present indicative.
Almost. Narrat has no "case", since case only applies to declining a noun. Narrare is a verb and has a conjugation. Every other word is a noun and therefore has a case. All nouns and adjectives and only nouns and adjectives have cases. The adjective takes on the same gender and case as the noun that it is modifying. It still does even if it stands by itself.
Poeta - Nominative Singular
Puellae - Dative Singular
Fabulam - Accusative Singular
Narrat - 3rd person singular present indicative
Edit: I saw someone provide a translation that is different than what I think it is. So to be very clear I'm providing my translation. "The poet tells a/the story to the girls."