This issue was, initially quite a problem for me. As a result of his fall, Satan was banished to earth. We don't know when Satan's fall occurred. If it was billions of years ago, then Satan could be responsible for billions of years of suffering and death. In this scenario Eden was the first piece of territory God took back from Satan, and Adam and Eve were to be the first of His army for taking the remainder of earth back.
Punishment is always most effective when administered immediately.
A third possibility is that human death is meant by the above passages. I'm not trying to say that God doesn't care about animals. I'm sure He loves them. If you don't accept that, then you have to presume a radically different ecology prior to the fall. I'm not saying it couldn't be, but the most reasonable approach IMO is to be willing to admit that just as we do not totally understand nature, we don't totally understand from Scripture the conditions that existed prior to the fall.