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I believe (though have not tested) that if you lose the other eye also, the Eye of Aiming will still give you the precision bonus, applied in this case to a base precision of 0.

Afflictions that are caused by an item can't be healed as long as the item is attached. Since most such items are cursed (not all - the Bane Venom Charm and Fever Fetish aren't) these afflictions are more or less permanent. Also, if a unit has an affliction "built in" (such as the Allfather) this affliction can't be healed.