Death, the cessation of life, and death, the forgetting or cessation or mortality of the immortal soul are two different things. The point in case here is the corruption and mortality of the immortal but passable soul. You can see Athanasius' 'On the Incarnation' about this. Humans are intended and made to be immortal. The immortality of the soul is of course not a form of self-existence but is so in connection with God, or as some say, 'immortal by grace'. So when you look at the death of say, a dog, it is not indicative of anything but that dogs die - it may be the case that dogs are made to die, but in any case, they do not seem to be aware of their mortality.
By the way, in patristic literature, this idea of cessation of life is connected with the bestial, or lowering of man from his proper rank to that of brute animals. So even early on you see that the death of plants, animals and so forth is not seen necessarily as a result of man's fall, but as the natural condition of such things, which man falls into in disobedience and separation from God. Thus while this may be natural for a plant or an animal, it is not so for man. Plants and some creatures also don't seem to be individual 'lives' in the same way people are; take for instance the grafting of plants together to form one seamless organism.
A second point, is that some think that the original creation, that of paradise, is entirely immortal just as man was immortal, but that in being cast our of paradise, man enters into a chaotic, death filled world, even as he himself becomes chaotic and death filled. This being so that he does not become as Satan is; disobedient, impassible and immortal.