If another effect (or effects) modifies how much damage a Wizard you control would deal—by preventing some of it, for example—the player being dealt damage or the controller of the permanent being dealt damage chooses the order in which any such effects (including Trance Kuja's) apply. If all of the damage is prevented, Trance Kuja's effect no longer applies.

The Wizard token's ability resolves before the spell that caused it to trigger. It resolves even if that spell is countered or otherwise leaves the stack.

If damage dealt by a Wizard you control is being divided or assigned among multiple permanents and/or players, that damage is divided or assigned before doubling. For example, if you attack with a 4/6 Wizard with trample and it's blocked by a 1/1 creature, you can assign 1 damage to the blocker and 3 damage to the defending player. Those amounts are then doubled to 2 and 6, respectively.

The damage is dealt by the same source as the original source of damage. The doubled damage isn't dealt by Trance Kuja unless it was the original source of damage.