Suppose I Dubious Challenge into two Noxious Gearhulk. Does my opponent need to put his trigger on the stack immediately when he puts his Noxious Gearhulk in play, which, reading sequentially, is before my Noxious Gearhulk is on the battlefield to be chosen as a target?
Does that mean if the battlefield is empty before Dubious Challenge is cast, generally I'll end up with a Noxious Gearhulk and my opponent will end up with nothing?
No. Triggers go on the stack before someone receives priority and no one receives priority during the casting of a spell.
Both Gearhulks will trigger and once Challenge is done resolving both triggers will go on the stack in APNAP order - yours first, then his. You declare a target (your opponent's gearhulk) then he declares a target (your gearhulk). After both triggers resolves both gearhulks will be in your graveyard and you'll both be up 4 life.
Triggered abilities can't be put on the stack during the resolution of a spell or ability; they trigger, but they're only put on the stack after the spell is done resolving, before the active player would get priority. And targets are only chosen when the abilities are put on the stack. So both Gearhulks would be on the battlefield at that point, and they can target each other. End result would be that both players gain 4 life.
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I'm a former judge (lapsed), who keeps up to date on rules and policy. Keep in mind that judges' answers aren't necessarily more valid than those of people who aren't judges; what matters is we can quote the rules to back up our answers. When in doubt, ask for such quotes.
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Does that mean if the battlefield is empty before Dubious Challenge is cast, generally I'll end up with a Noxious Gearhulk and my opponent will end up with nothing?
Both Gearhulks will trigger and once Challenge is done resolving both triggers will go on the stack in APNAP order - yours first, then his. You declare a target (your opponent's gearhulk) then he declares a target (your gearhulk). After both triggers resolves both gearhulks will be in your graveyard and you'll both be up 4 life.