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The copy 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).

The ability granted by Dual Casting can target (and copy) any instant or sorcery spell you control, not just one with targets.

You can’t choose to pay any additional costs for the copy. 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 copy too. For example, if you sacrifice a 3/3 creature to cast Fling and then copy it with the ability granted by Dual Casting, the copy of Fling will also deal 3 damage to its target.

If the spell being copied has an X whose value was determined as it was cast (like Earthquake does), the copy has the same value of X.

If the spell being copied is modal (that is, it says “Choose one —” or the like), the copy will have the same mode(s). You can’t choose different ones.

The copy is created on the stack, so it’s not “cast.” Abilities that trigger when a player casts a spell won’t trigger. The copy will then resolve like a normal spell, after players get a chance to cast spells and activate abilities.