In the Coen's brother movie Inside Llewyn Davis, does the red cat have a significance? Is it just a narrative expedient or does it have a particular meaning?
From a January 2014 Time Out interview:
Dave Calhoun: A runaway ginger tom has a big role in the film. You’ve said that you only included him to help glue a wayward story together. Is that a joke?
Joel Coen: A half joke. Unlike a lot of the stuff we’ve done, there isn’t a plot. The question was: what gives the movie its momentum? We were like: he’s got a cat. What happens to the cat? So there’s another thing you follow. Also, we’re telling a story about a character who has a difficult time with humans but has to relate to an animal. That reveals things.
In "Inside Llewyn Davis": America at its ugliest, Eileen Jones describes the role of the cat:
The cat is very important to the Llewyn Davis narrative. A handsome, expressive orange tabby that escapes from the apartment where Llewyn Davis is crashing, the cat becomes a minor obsession of Llewyn’s. He keeps losing and finding it, chasing and carrying it around with him. But for all his trouble, the cat he returns to its worried owners turns out to be a female orange tabby virtually identical to the cat he lost. “Where’s his scrotum?” shrills Mrs. Goldfein. In valiantly trying to safeguard the cat, or rather both cats, Llewyn endeavors to get one small symbolic aspect of his life under control. And fails.
The circular storyline of the cat is part of the overall relentless cycle of the narrative, which starts and ends at the same place, in an alley outside a folk music club where Llewyn Davis is getting beaten up. This beginning/ending scene was the inspiration for the film, according to Ethan Coen: “We were in the office, and Joel said, ‘OK, suppose Dave Van Ronk gets beat up outside of Gerde’s Folk City. That’s the beginning of a movie.’”
Speaking about Van Ronk, here is a link to the cover of his 1964 album Inside Dave Van Ronk. A cat is in the lower right corner.
If you want to know more about how the cats were cast, see this Collider interview with Christina Rash.
The conclusion is that the cat is mostly a narrative expedient since, as Joel Coen noted, there isn't really a plot. Since Llewyn Davis is to some extent based on Dave Van Ronk, considering the Inside Dave Van Ronk album cover, it made sense to include some cat in the first place.