I wouldn't say he enjoys killing (credits) he just doesn't hold the kind of morale value as "even scum deserves to live".
He's a martial artist from head to toe, if people are trying to kill him, he won't think twice about killing his opponents.
He threatened to kill her so she'd get scared and flee from him (which isn't really the brightest idea he could have had, since she's got an "impressive" background, scaring her away could turn as a bad outcome for him, he could just have her get bored of following him). I don't think he'd kill someone for small reasons however, if he has a good reason to kill someone (such as being life-threatened like most his encounters up to now) he won't hesitate returning the favor.