Is a nature possible where animals do not kill each other?
No, even herbivores will eat other animals, and predation goes down ot the cellular level.  Also something has to be ating your dead and will have a huge evolutionary pressure to evolve toward predation with no predators present. 

Animals by definition have a much better profile of nutrients for other animals.

Deer routinely kill and eat birds, eggs and even dead rabbits. In nature food is food and evolution does not care where you get it from. Carnivores will and have evolved from an herbivorous ancestor when no carnivores are present.