Why does killing produce bad karma? Is it because we cause a being to suffer and experience pain while it's/he's being killed?

Or, is it because we put an end to its/his relationships and desires?

Or what reason can it be?
It is our Perception of the Sensation / Feeling experienced when result bears fruit is what distinguished the Karma as bad and not. I.e., Bad Karma is what you Perceive the results as undesirable and Good Karma is what we would perceive the results as favourable when the doer experiences the result, as well as skill full when this is conducive to reach the Buddhist Goal of Nirvana and un skilful when it is un conducive in reaching the final goal.

Say you kill you will experience the same when your actions catcher upto you. Hence what is needed is to treat other like you would be treated, or treatments you would evaluate positively if you experience it. 

Also there is skilfulness of your action towards the goal of liberation. In this light any action which creates fabrication which hamper concentration and insight or the opportunity to practice can be considered unwholesome. Killing has aversion and / or craving as base. This creates fabrication which give painful sensation when meditating which will hamper concentration, insight as well as the needed longevity and conducive human birth to practice Vipassana hence such actions are un skilful.

For a worldly person (or someone not very deep and intensive in Vipassana) it is helpful to build-up Perception (Sanna) what these are good action and bad action so he is steered away from un skilful / un wholesome action towards skilful / wholesome action. As Perception clouds your sense of reality such mental maps ultimately have to be dissolved away too.