If it's not battle phase will a monster that looses -1 -1 health by an ability not spell. Will it kill it if it's only a 1/1 creature? Ex. Like the card korozda gorgon
Yes. A creature with 0 toughness can not live no matter when it is
(Neg for this?)
704.5f (http://imtgapp.com/forum/index.php?action=imtg;area=rule;number=704.5f): If a creature has toughness 0 or less, it's put into its owner's graveyard. Regeneration can't replace this event.
This is a state based action and resolves the next time a player would receive priority.