If I putLeveler onto the battlefield, Can I flash a Donate in response to the trigger to give the permanent to an opponent? Can I use Zedruu the Greathearted's mana ability in response to the trigger and give Leveler to an opponent without leveler exiling my library?
605.1a An activated ability is a mana ability if it meets all of the following criteria: it doesn’t
require a target (see rule 115.6), it could add mana to a player’s mana pool when it resolves, and
it’s not a loyalty ability. (See rule 606, “Loyalty Abilities.”)
No. You control the ability, so you'll be the one following its instructions.
603.3a A triggered ability is controlled by the player who controlled its source at the time it
triggered, unless it’s a delayed triggered ability. To determine the controller of a delayed
triggered ability, see rules 603.7d–f.
608.2c The controller of the spell or ability follows its instructions in the order written. However,
replacement effects may modify these actions. In some cases, later text on the card may modify
the meaning of earlier text (for example, “Destroy target creature. It can’t be regenerated” or
“Counter target spell. If that spell is countered this way, put it on top of its owner’s library
instead of into its owner’s graveyard.”) Don’t just apply effects step by step without thinking in
these cases—read the whole text and apply the rules of English to the text.
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Why bother with mere rulings when so many answers can be found in the Rules?
Leveler's ability triggers immediately as Leveler enters the battlefield. You get to put the ability on the stack before receiving priority to cast a spell or activate an ability in response.
Even if you give Leveler away in response, its ability is still on the stack under your control, so you still has to exile your library when it resolves even if you don't control Leveler anymore.
(As an aside, Zedruu's activated ability isn't a mana ability. Mana abilities are those that add mana).
Donate is a sorcery, so you would need something like Vedalken Orrery. (And this probably won't have the result you want.)
That card doesn't have a mana ability.
Yes.
No. You control the ability, so you'll be the one following its instructions.
Even if you give Leveler away in response, its ability is still on the stack under your control, so you still has to exile your library when it resolves even if you don't control Leveler anymore.
(As an aside, Zedruu's activated ability isn't a mana ability. Mana abilities are those that add mana).
It sounds like you have an old copy of the rules. I don't think this answer is affected by any of the changes since then, but you might want to get a more recent copy: https://magic.wizards.com/en/game-info/gameplay/rules-and-formats/rules