Can I win with a "you lose" effect on the stack?
In general, you could win while a "you lose" effect is on the stack. Winning/losing is a state-based action and the stack doesn't finish resolving if the game is over (e.g. a 2 player game).

However, with the Immortal Coil situation you describe, you lose. This is because the Immortal Coil in question won't create another triggered ability on the stack until the first one isn't there.


  603.8. Some triggered abilities trigger when a game state (such as a player controlling no permanents of a particular card type) is true, rather than triggering when an event occurs. These abilities trigger as soon as the game state matches the condition. They'll go onto the stack at the next available opportunity. These are called state triggers. (Note that state triggers aren't the same as state-based actions.) A state-triggered ability doesn't trigger again until the ability has resolved, has been countered, or has otherwise left the stack. Then, if the object with the ability is still in the same zone and the game state still matches its trigger condition, the ability will trigger again.


So the Immortal Coil can't trigger for your opponent until you've already lost the game, unless you're otherwise able to prevent or counter the ability on the stack.

Here's the bit I'm not certain about and will look up, but I think if you could flicker the Immortal Coil once the opponent controls it, it will trigger, because the "new" Immortal Coil has no memory of being attached to the previous ability (still on the stack) and thus will create a new one.