A copy is created even if the spell that caused the ability to trigger has been countered by the time that ability resolves. The copy resolves before the original spell.

You can't choose to pay any additional costs for the copy created by Jin-Gitaxias's first ability. 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.

If the spell that's copied is modal (that is, it says “Choose one —” or the like), the copy created by Jin-Gitaxias's first ability will have the same mode or modes. You can't choose different ones.

If the spell that's copied had damage divided to targets as it was cast, the division can't be changed, although the targets receiving that damage still can. The same is true of spells that distribute counters to targets as it was cast.

Jin-Gitaxias's last ability triggers once each turn, not once each turn per opponent.

The copies that Jin-Gitaxias's first ability creates are created on the stack, so they're not “cast.” Abilities that trigger when a player casts a spell won't trigger.