So that mass panic in the theater would have happened anyway, without anyone shouting "fire"?
Perhaps not, but the supposed "crime" was falsely shouting "fire", and the panic would have happened even if there really was a fire. In any case, no one forced the other patrons to panic. If they did, it was entirely their own fault, and they—not whoever shouted "fire"—are wholly responsible for the consequences.
Looking at the events after they're happened, we can conclude that someone unnecessarily shouting "fire" in a crowded theater was a sufficient condition (thus, leading) to people getting hurt. Was it a necessary condition for people getting hurt that way? Common sense says it was.. also anyone who's looking someone to blame.
Wrong on both counts. It is neither a sufficient condition, since people could simply refrain from panicking despite the (maybe false) warning, nor a necessary one, as there could be a panic even if the warning was true, or for that matter without anyone shouting "fire" at all.
If you offer an incentive (survive a theatre fire / money) to someone for committing a crime (trampling someone to death / shooting someone), did you cause it, or was it all on the person who committed the act?
Obviously, the person who committed the act. Otherwise one would be forced into the absurd conclusion that property owners are at least partly responsible for the theft of their property, since simply by having it they give potential thieves an incentive to steal. The same could be said of many other crimes. Responsibility rests with the one whose choices led directly to the outcome. Merely offering an incentive does not make one responsible for the consequences of someone else's choice.
If you manipulate someone by giving them false information, that could be said to take away their ability to choose freely, making you responsible for the consequences when they make the right choice given the information you provided and harm results because it was false. However, that does not apply here, because trampling others in one's haste to escape is something one would be responsible for even if the fire were real; the harm was not due to the information being false.