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 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. A different mode can't be chosen.

If the spell that's copied has an X whose value was determined as it was cast (like Kozilek's Command does), the copy will have the same value of X.

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.

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