Izzet Steam Maze

The controller of the copy 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.

If an ability of a plane refers to "you," it's referring to whoever the plane's controller is at the time, not to the player that started the game with that plane card in their deck. Many abilities of plane cards affect all players, while many others affect only the planar controller, so read each ability carefully.

The copy that Izzet Steam Maze's first ability creates is created on the stack, so it's not "cast." Abilities that trigger when a player casts a spell (like Izzet Steam Maze's first ability itself) won't trigger.

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

A face-up plane card that's turned face down becomes a new object with no relation to its previous existence. In particular, it loses all counters it may have had.

The effect of Izzet Steam Maze's first ability is mandatory. You must copy the spell whether you want to or not.