The copies storm creates are created on the stack, so they're not cast. Abilities that trigger when a player casts a spell (such as storm) won't trigger.

Storm counts spells cast before the spell with storm was cast. Spells cast after the spell with storm was cast but before the storm ability resolves aren't counted.

The storm ability and the copies it creates all resolve before the spell with storm. They resolve even if the spell with storm is countered.