Ewan MacColl's The Shoals of Herring was composed for and first broadcasted in a BBC 'Radio Ballad' in August 1960. Since the film is set in 1961, Llewyn Davis could not have recorded this as an eight year old.
While driving to Chicago, the "valet" Johnny Five opens a conversation by muttering "Orlovsky" and "Clean Asshole Poems." That would be Peter Orlovsky, who was Allen Ginsburg's long-time partner. But "Clean Asshole Poems" wasn't published until 1978, and the movie is set in 1961.
Llewyn Davis uses two different capos. In the clubs it appears to be a Hamilton capo which was around in the 1960s (although possibly not as early as 1961) but in other scenes he is clearly using a Shubb capo which was not available until 1980.
Early in the film, Llewyn takes the subway downtown to Greenwich Village, and when he arrives at Sheridan Square, we see him coming up from the uptown platform, as if he had been traveling from the other direction. (He should have come up from the staircase across the street visible in the same shot.)
When the driver is arrested, the camera zooms to the ignition switch of the 1961 Buick. the keys are gone, but on that model the car would still start and run by turning the switch without a key. in the run position the key can be removed. in the lock position-full left-it cannot be restarted.
After Llewyn returns from his Chicago trip and is staying at his sister's house, the Etch-A-Sketch by the bed reads "Welcome Uncle Llewyn," written in capital block letters (except for the "U" which connects to the "N"). However, it is not possible to create unconnected patterns on an Etch A Sketch screen, as the stylus only moves in a connected line, so only cursive(script) style of writing is possible.