When Ed visits Doris in prison, a photograph of a circle is seen on the table in front of him - a link both to Circle Films, which was the company behind the Coen brothers in their earlier films, and The Hudsucker Proxy (1994)'s circle gag.
Joel Coen and Ethan Coen came up with the story while working on The Hudsucker Proxy (1994). While filming the scene in the barbershop, the Coens saw a prop poster of 1940s haircuts and began developing a story about the barber who cut the hair in the poster.
When Ed Crane is at the French piano teacher's studio (Jacques Carcanogues), the actor portraying him, Adam Alexi-Malle, plays the opening of the Piano Concerto #1 by Franz Liszt. This was unplanned and spontaneous during one of the takes during shooting as Joel Coen and Ethan Coen were unaware that Adam was in fact an accomplished concert pianist.
Billy Bob Thornton jokingly made it look like Ed Crane had an erection in one of the scenes where he's watching Scarlett Johansson's character playing the piano. Only one of the prop guys noticed during production. When the Coen Brothers later found out, they made it clear that Ed would not be aroused in the scene.
In his autobiography "The Billy Bob Tapes", Billy Bob Thornton said that he had the flu during the last two weeks of shooting. When the barbershop scenes were shot during that period, Thornton said you may notice in the scenes that his symptoms appear with puffy eyes and he's stuffed up.