The copy is created on the stack, so it's not "cast." Abilities that trigger when a player casts a spell won't trigger.

Izzet Guildmage's abilities can copy any instant or sorcery spell with an appropriate mana value, not just one with targets.

If a spell has {X} in its mana cost, include the value chosen for that X when determining the mana value of that spell.

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).

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

If the spell has damage divided 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.

The controller of a copy can't choose to pay any alternative or additional costs for the copy. However, effects based on any alternative or additional costs that were paid for the original spell are copied as though those same costs were paid for the copy.

Any choices made as a spell resolves won't have been made yet once it's copied. Any such choices will be made separately as the copy resolves.

If you copy a spell, you control the copy. It will resolve before the original spell does.