I control Willbreaker and my opponent controls a random creature (let's be simple and call it a Fire Elemental). If I cast Chain of Plasma targeting the Fire Elemental, Willbreaker should trigger and its ability should go on the stack and resolve above Chain of Plasma. Does this mean that I'm the creature's controller by the time Chain of Plasma checks, and that I am the one given the option to discard a card to copy Chain of Plasma? When I "choose a new target" for the copy of Chain of Plasma, that should trigger Willbreaker again (assuming I target an opponent's creature) and let me repeat the process, right?
The answer shouldn't be any different if I target an opponent's creature with lower toughness (I.e. something that dies to Chain of Plasma's damage), right?
Yes, this works as you described. Chain of Plasma checks the creature's controller when it resolves, so since you gain control of that creature with Willbreaker before the Chain resolves, you get the option to discard a card and copy the Chain for another round, and so on, until you run out of cards to discard, run out of creatures to target, or choose not to copy the Chain anymore.
Even if the Chain's damage is lethal to the creature, it dies as a state based action after the copy of the Chain has already been created. And even if it would die right away, the game would just use last known information to determine that creature's last controller.
In this scenario, Chain of Plasma checks which player controls the targeted permanent (C.R. 108.1, 115.4) when that spell resolves; that player is not necessarily the one who controlled that permanent when Chain of Plasma (or a copy of it) was put on the stack (C.R. 608.2h). Thus it's that player (that is, the one who now controls the targeted permanent) who chooses whether to discard a card with Chain of Plasma.
EDIT (Jun. 8, 2021): Edited to conform to updates in Chain of Plasma and the rules in the meantime.
The answer shouldn't be any different if I target an opponent's creature with lower toughness (I.e. something that dies to Chain of Plasma's damage), right?
Even if the Chain's damage is lethal to the creature, it dies as a state based action after the copy of the Chain has already been created. And even if it would die right away, the game would just use last known information to determine that creature's last controller.
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EDIT (Jun. 8, 2021): Edited to conform to updates in Chain of Plasma and the rules in the meantime.