My opponent MUST put their basic down first before they draw 1 card for my mulligan, right? Say my opponent draws a better choice starter Pokémon when I mulligan, are they allowed to switch it out with another from their hand?
They must place their active Pokémon and their prizes before optionally drawing a card for the mulligan. Once placed, the active cannot be changed during setup; but you are allowed to add a new Pokémon to your bench, if there is room.
Source: PUI Rules Team (2005-06-16), PUI Rules Team (2006-02-09), PUI Rules Team (2006-07-27)