If I use Arceus' "Type Switch" attack and choose a type, what happens if I use Ninja Boy to switch Arceus for a Basic Pokemon of a different type? Does the new Pokemon become both types or does the new type supersede the choice from "Type Switch"? Or what if you Ninja Boy for a dual-type Pokemon like Volcanion EX that is Fire/Water?

The type chosen by Type Switch overrides the Pokemon's actual type, so the new Pokemon would become the type chosen rather than its regular type. And in the case of a dual-type Pokemon it would become the single type that you chose for Type Switch.

Relevant cards

XYP_XY116

XYP_XY197