Why does the left-handed twin choose a "mere stick" to fight the duel, while the right-handed twin selects a deer antler?

This was because they both already knew that the right handed twin who was the personification of good will kill the left handed twin who is a representation of evil.

According to the myth "The World on Turtle's Back", after killing the left handed twin, the right handed twin threw him off the Earth causing him to rule the underworld. The story is a creation myth which explains the creation of moon, seas, underworld and much more.