In Adventure Time's Card Wars game (the physical card game, not the computer game), can I play a 0 action spell on my opponent's turn?
No, this is explicitly forbidden in rules
According to section "Playing cards → Rainbow cards" of the rule book (page 5):
Creatures, Buildings, and Spells with an Action cost of 0 are always Rainbow cards. They may only be played during your own turn (even after spending your 2 Actions), but require no Landscape types to play.
Link to the rule book is taken from this page