If Orim's Thunder is kicked, but the artifact or enchantment is an illegal target as Orim's Thunder tries to resolve, Orim's Thunder won't be able to determine the artifact or enchantment's mana value. No damage will be dealt to the target creature, even if it remained a legal target. If only the creature has become an illegal target, the artifact or enchantment will still be destroyed. (If both targets become illegal, Orim's Thunder won't resolve at all.)

If Orim's Thunder is kicked, it's the source of damage dealt to the target creature, not the artifact or enchantment.

If a spell's kicker cost was paid, the spell is "kicked."

The kicker ability doesn't let you pay a kicker cost more than once.

If you copy a kicked spell on the stack, the copy is also kicked. If the copied spell is a permanent spell, the token the copy of that spell becomes when it enters is also kicked.

If a card or token enters as a copy of a permanent, the new permanent isn't kicked, even if the original was.

To determine a spell's total cost, start with the mana cost (or an alternative cost if another card's effect allows you to pay one instead), add any cost increases (such as kicker), then apply any cost reductions. The spell's mana value remains unchanged, no matter what the total cost to cast it was.

If you put a permanent with a kicker ability onto the battlefield without casting it, you can't kick it.