Why is death a vital piece of God's plan in LDS theology?

Necessity of the Spirit World

In LDS theology, the Spirit World is the place where spirits reside between the time of death and the resurrection. Our time there plays an important role in our salvation, especially due to the doctrine of Salvation for the Dead.

Death is the separation of body and spirit.

During our life here, our bodies are mortal, corruptible, and subject to physical disease and pain. This is a consequence of the Fall, and part of the reason for it is so that we can learn from experiences unique to mortality.

Because of the atonement of Jesus Christ, which includes his resurrection, all people qualify to be resurrected, and will receive resurrected bodies, which are immortal, free from physical imperfections, and glorified (to varying levels).

Our spirits can't receive a new body until they leave our old body. We call this death.

I think there's some confusion in your question about the spirit world and about translated beings.

During the time that our spirits are separated from our mortal bodies, after death, we wait in the spirit world for our resurrection and the final judgement. Depending on certain factors, we may be assigned to spirit prison or spirit paradise, within the spirit world (though these are not final assignments, because the final judgement doesn't happen until we're ready to be resurrected).

Only spirits without bodies can be in the spirit world (which is why Jesus visited the spirit world during the time between his death and resurrection). Translated beings do not go to the spirit world, because translated beings still have bodies. Their mortal bodies are made temporarily immortal and they receive some measure of protection and power.

Even those who are translated will die – that is, at some point, they will receive their final judgement and their spirits will leave their translated bodies to receive perfected, glorified, resurrected bodies – perhaps "in the twinkling of an eye."

Wow, this is a good question. I dove into the search tools on LDS.org and came up with a few things that you've probably seen already, but they were instructive for me.

It looks like the reason for death has directly to do with circumstances of the Fall.

In Alma 12:23-24, it says:

23 And now behold, I say unto you that if it had been possible for Adam to have partaken of the fruit of the tree of life at that time, there would have been no death, and the word would have been void, making God a liar, for he said: If thou eat thou shalt surely die.

24 And we see that death comes upon mankind, yea, the death which has been spoken of by Amulek, which is the temporal death; nevertheless there was a space granted unto man in which he might repent; therefore this life became a probationary state; a time to prepare to meet God; a time to prepare for that endless state which has been spoken of by us, which is after the resurrection of the dead.

So death must come, among other reasons, because God said so. Is there yet a reason behind that? I don't know. There's certainly lots to learn from death that may only be accomplished by having death. For instance, how does knowing that we will die affect how we live?

Or, would it fair to say, that since Adam and Eve became even as gods through the Fall, knowing good from evil, and comprehending opposites, that life/birth must have its opposite, being death?

We've also got Alma 22:14:

14 And since man had fallen he could not merit anything of himself; but the sufferings and death of Christ atone for their sins, through faith and repentance, and so forth; and that he breaketh the bands of death, that the grave shall have no victory, and that the sting of death should be swallowed up in the hopes of glory; and Aaron did expound all these things unto the king.

Although we aren't all translated, if we're righteous, the sting of death is swallowed up in Christ (1 Cor. describes the sting as sin).

Besides, if all people were just translated/taken up instead, I think that kind of death wouldn't cause us to search for something more to life. Dying has a purpose in urging us to consider our state of being and our standing with God, or even to search for God, while we yet live. This thought is mostly philosophical, but even prophets use this reasoning to convince us to "consider our ways," (Haggai 1:5) and "turn again to the Lord" (Lamentations 3:40).