I know you can't play once per turn effects "at the beginning of the game," but can you play "once per game" effects during that phase?

It all depends on what kind of specific occurence we're talking about. What power/trait are you asking about because it is highly likely that the answer changes depending on which specific one

Let's use this as our baseline

Professor X


FORGING THE CEREBRO SWORD
Once per game when an opposing equipment would be KO'd, you may instead equip it to a friendly character. If you do, that character can use Blades/Claws/Fangs this game.


Opposing character replaces an equipped figure. Equipment gets unequipped, and triggers an "unequip: drop" effect. Can you use your once per game right then to equip the equipment

Forging the Cerebro Sword only triggers on KOs, not drops. If for some reason your opponent swapped out a figure equipped with an Unequip: KO item, you could grab that.

Sorry, yes, that's what I meant

Unequip: ko, and the once per game would be allowed. Thanks