Humanity is screwed, but that's good.
Don't worry - we wouldn't want to throw ourselves in with this race evaluating humanity. If you judge by its tests, it doesn't have nice motives in general.
The button presser isn't killing the other person using the button - the other person dying via gun is causing the button presser to press the button.
The trigger is in the gun - there's a transmitter that goes off when the trigger is pulled. So when the trigger is pulled, the box sends of a psychic signal making the nearest person press it.