Short answers, pulled from the story.
Melancholia was written and directed by Lars von Trier. The idea originated during a therapy session he attended while being treated for depression, after a therapist told him that depressive people tend to act more calmly under extreme pressure because they already expect the worst.
Kirsten Dunst won the Best Actress Award at the 64th Cannes Film Festival, where Melancholia premiered on the 18th of May 2011. She also won Best Actress at the European Film Awards, the Los Angeles Film Critics Association, the New York Film Critics Circle, the National Society of Film Critics, and the Saturn Awards, among many others.
The Depression Trilogy is an unofficial grouping of three films by Lars von Trier. Melancholia is the second entry, preceded by Antichrist in 2009 and followed by Nymphomaniac in 2013.
The prelude to Richard Wagner's opera Tristan und Isolde, composed between 1857 and 1859, serves as the film's main musical theme. Von Trier chose it after reading a passage in Marcel Proust's In Search of Lost Time in which Proust calls Wagner's prelude the greatest work of art of all time.
Principal photography took place in Sweden from the 22nd of July to the 8th of September 2010. Interior scenes were shot at Film i Väst studios in Trollhättan, and exterior scenes were filmed at and around Tjolöholm Castle in the Halland region.
At the post-screening press conference, director Lars von Trier made jokes about Nazis, said he understood Adolf Hitler, and expressed admiration for architect Albert Speer, jokingly announcing he was a Nazi. The Cannes Film Festival issued an official apology the same day and declared von Trier persona non grata the following day, barring him from within 100 meters of the Festival Palace.