Short answers, pulled from the story.
My Neighbor Totoro was written and directed by Hayao Miyazaki and released in 1988. It was produced by Studio Ghibli for Tokuma Shoten and opened on the same bill as Isao Takahata's Grave of the Fireflies.
My Neighbor Totoro won the Kinema Junpo Award for Best Film, the Mainichi Film Award, the Animage Anime Grand Prix prize, and the Special Award at the Blue Ribbon Awards, all in 1989. In 2022, the stage adaptation at the Barbican Centre won six Olivier Awards.
Totoro is identified in the film as mori no nushi, or master of the forest, and is regarded by scholars as a kami - a Shinto spiritual entity. He lives in a camphor tree within a Shinto shrine, and Mei's father takes both daughters to greet and thank him after she encounters him, mirroring standard Shinto practice.
The Totoro Hometown Fund Campaign is a Japanese conservation effort that used the Totoro character as its mascot to preserve satoyama landscapes in Saitama Prefecture. Started in 1990, it held an auction in August 2008 at Pixar Animation Studios selling more than 210 original artworks inspired by the film.
Joe Hisaishi composed the score, hiring a full orchestra and using a Fairlight instrument as a central element. The soundtrack was first released in Japan on the 1st of May, 1988, by Tokuma Shoten.
A thirteen-minute sequel written and directed by Miyazaki exists but has never been released on home video. It follows Mei through a night of adventures with Kittenbus and has been shown regularly at the Ghibli Museum in Japan since its 2003 debut.