There are no logical reasons that killing people and animals is bad.
Every moral rule comes down to a core belief that can't be proven. It has to be taken on faith. For example, your reasons are based on the fact that people feeling pain or sorrow is bad (something that you haven't proven).
My core moral belief is that human life is sacred. My other morals (which include not killing people) stem from that.
But I have no logical way to prove that human life is sacred. I just believe it.