The diamond pattern on the sword's handle is not seen for most of the episode.
When Jack's father is riding the horse, you see the moon is a cresent with a star within its shape. The star which the cresent surrounds shouldn't be there unless the star was in front of the moon (which is impossible) or the moon is actually a cresent (in which case, it wouldn't be able to orbit the planet).
Genndy Tartakovsky was nominated for the 2004 Annie Award for "Outstanding Directing in an Animated Television Production" for this episode. Scott Wills won the 2004 Annie Award for "Outstanding Production Design in an Animated Television Production" for this episode. Genndy Tartakovsky, Brian A. Miller, Don Shank, Robert Alvarez, Randy Myers, Mun Jeong Yu, Bong Koh Jae and James T. Walker won the 2004 Emmy Award for "Outstanding Animated Program (For Programming Less Than One Hour)" for this episode.
The white stallion: Mononoke-hime - Forest God.
When the mythical cloud horse (possibly Slephnir) comes down to help Jack's father out of a tight spot, his hooves--when connecting to the ground--make flowers, vines, and grass spring from dead earth. A similar scene is in Hayao Miyazaki's Mononoke-hime (U.S. Title: Princess Mononoke) as the Forest God walks, things grow.
Jack's father rides a cloud, just like the Monkey King, hero of the perennially popular East Asian folktale Journey to the West. The Monkey King Reference is a huge myth in Asian culture. It can be seen everywhere. Among other things it is the basis for Dragonball Z.