The copy is a spell, so it can be targeted by other spells and abilities that target a spell.

The player putting the copy spell on the stack controls that spell.

The spell copy has all the text and can result in itself being copied.

As Chain Lightning resolves, the targeted player or the controller of the targeted creature may copy it. The copy has the same text, target, and color of the resolving spell, though the player who’s copying it may choose a new target for it. Once that copy is created (or not), the first Chain Lightning has finished resolving and leaves the stack.

The player putting the copy of the spell on the stack controls that copy.

The copy of Chain Lightning is created on the stack, so it’s not cast. Abilities that trigger when a player casts a spell won’t trigger. Players may respond to that spell before it resolves.

If the targeted creature or player is an illegal target when Chain Lightning tries to resolve, the spell is countered. It can’t be copied.