Archetypes and Eponymous Names
As a rule, you can't simply drop the definite or indefinite articles when they're required. There are other great answers here that address the standard grammar rules involved, but there's an edge case that no one's covered yet: fables, fairy tales, allegories, and myths can sometimes feature eponymous characters where the noun is actually a proper name. This is especially common in Native American stories and myths. Consider this example:
Coyote is a bad hunter who never kills anything. Once he watched Eagle hunting rabbits, catching one after another - more rabbits than he could eat.
Coyote thought, "I'll team up with Eagle so I can have enough meat." Coyote is always up to something.
In this art form, Coyote and Eagle are proper names used by the story's archetypes. Because they are the names of specific characters, they are not preceded by a definite or indefinite article.
Most native speakers will not generally notice high frequencies of "the" in a well-formed sentence (see The Cat in the Hat for a sterling example), but will definitely notice if it's routinely missing. If you're concerned about too much repetition of the article + noun, though, you should replace it with a pronoun (e.g. he, she, it, they) when the antecedent is clear, or give your characters proper names after introducing them in your story. As a somewhat contrived example:
A snowman met a fish in the river. The fish was a trout. "Salutations, Trout," said the snowman.
"Hello, snowman," replied Trout.
"Let me introduce myself properly. I'm Frosty."
"Nice to meet you, Frosty," Trout responded politely.
Based on the story's progression:
- "A fish" => "the fish" as the story gains specificity.
- "A trout => "Trout" as a proper name or form of address (not something you'd likely see outside of fiction writing) because a character gave him an eponymous name.
- "Snowman" as a term of address, which doesn't need an article => "Frosty" which is the snowman's proper name and therefore doesn't need an article.
Unless you're using it incorrectly, no one will think "the fish" or "the snowman" is redundant, but stylistically you should definitely explore ways to vary your descriptions and attributions throughout the story for artistic effect. Just don't drop the articles willy-nilly.