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Ahoi,
ich versuche mich mal an der Lösung der aufgeworfenen Fragen 
Bzgl. 1.): 800.4a When a player leaves the game, all objects (see rule 109) owned by that player leave the game and any effects which give that player control of any objects or players end. Then, if that player controlled any objects on the stack not represented by cards, those objects cease to exist. Then, if there are any objects still controlled by that player, those objects are exiled. This is not a state-based action. It happens as soon as the player leaves the game. If the player who left the game had priority at the time they left, priority passes to the next player in turn order who’s still in the game.
Das klingt für mich so, als dass die Krone schlicht exiliert wird.
Bzgl 2.) 603.6c Leaves-the-battlefield abilities trigger when a permanent moves from the battlefield to another zone, or when a phased-in permanent leaves the game because its owner leaves the game. These are written as, but aren’t limited to, “When [this object] leaves the battlefield, . . .” or “Whenever [something] is put into a graveyard from the battlefield, . . . .” (See also rule 603.10.) An ability that attempts to do something to the card that left the battlefield checks for it only in the first zone that it went to. An ability that triggers when a card is put into a certain zone “from anywhere” is never treated as a leaves-the-battlefield ability, even if an object is put into that zone from the battlefield.
Demnach würdest du durch den Trigger sterben.
Bzgl 3.) Allein aus dem Bauch heraus hätte ich gesagt, dass das zwei "lose life" effekte sind, mithin zwei separate card-draw-Trigger in der Höhe von jeweils einer Karte, sodass der Alms Collector nicht betroffen ist. Aber wie gesagt, dass ist nur meine Intuition die hier spricht.
Grüße
