Each of the copies will have the same targets as the spell it's copying unless you choose new ones. You may change any number of the targets, including all of them or none of them. If, for one of the targets, you can't choose a new legal target, then it remains unchanged (even if the current target is illegal). If there are multiple copies, you may change the targets of each of them to different legal targets.

If the spell Storm King's Thunder copies has an X whose value was determined as it was cast (like Fireball does), the copy has the same value of X.

You can't choose to pay any additional costs for the copies. However, effects based on any additional costs that were paid for the original spell are copied as though those same costs were paid for the copies too. For example, if you sacrifice a 3/3 creature to cast Fling and then copy it with Storm King's Thunder, each copy of Fling will also deal 3 damage to its target.

If the spell Storm King's Thunder's ability copies is modal (that is, it has a bulleted list of choices), the copies will have the same mode(s). You can't choose different ones.

When the delayed triggered ability of Storm King's Thunder resolves, it creates X copies of a spell. You control each of the copies. Those copies are created on the stack, so they're not "cast." Abilities that trigger when a player casts a spell won't trigger. The copies will then resolve like normal spells, after players get a chance to cast spells and activate abilities.

Storm King's Thunder will create copies of the spell that caused its delayed triggered ability to trigger even if that spell has been countered by the time that ability resolves.