The recounting of the myth of Persephone is incorrect. She was not condemned to live in the Underworld for the rest of her life, but to spend part of the year with her mother Demeter, and the rest of the year with Hades in the underworld. As Demeter controlled the harvest, this myth was used to explain why the autumn and winter months are barren.
The plot centers on Reign digging a hole through the "center" of the earth, and coming out the other side, which is revealed to be near Madagascar. The continental United States and Madagascar are not antipodes (points opposite each other on a globe). The antipode of Madagascar is in the Pacific Ocean, about 500 miles away from Hilo, HI.
When Kara quotes Han Solo from Star Wars: Episode IV - A New Hope (1977), she's surprised Brainy doesn't recognize the reference. Considering what a cultural phenomenon the film was in 1977, and even still was in 2017 when he left Earth - exponentially more so than Bon Jovi's music - it seems odd that, having lived in the 31st century for seven years, Mon-El wouldn't have at least mentioned it to the Legion.