For a kill to count towards the Banestrike achievements, does it need to be primary damage amount like XP, or any damage at all like faction?
Grey mobs that don't give any XP will still count towards the various slayer achievements. The achievement window will show what your current kill count is in a particular category.
I've had grouped characters do absolutely nothing to mobs and still get the achievements, but only sometimes. Never figured out what caused it to count or not.
If you would normally get xp from the kill (not counting killing of very trivial mobs), you'll get slayer credit. I've had characters in a raid that kills stuff doing nothing but standing there picking their nose get credit for the kill. Unlike the hunter tasks, they do need to be in the same zone. I don't know if there's a distance restriction.