It is my understanding that there is a Savior for each world. As you have pointed out in an earlier answer, there is an Adam and Eve for every world. Clearly, there must be a Satan as well for each world.
Let me reiterate that Jesus Christ is both the Creator (under the direction of the Father) and the Savior of all the worlds. To demonstrate this we will take our information directly from the scriptures.
To demonstrate that He is the Creator of all the worlds, we will cite Moses 7:30,
And were it possible that man could number the particles of the earth, yea, millions of earths like this, it would not be a beginning to the number of thy creations; and thy curtains are stretched out still; and yet thou art there, and thy bosom is there…
This is one of the best non-technical definitions of infinity that I have seen. Here, Enoch is talking to with the Lord, and so we learn from this scripture that the Lord (Jehovah, Jesus Christ) is the creator of unlimited worlds.
In Moses 7:53 the Savior identifies himself as the Messiah, the King of Zion, the Rock of Heaven, which is broad as eternity. This scripture gives to his name the breadth of eternity–far beyond the confines of this earth.
In verse 56 we read the following:
And he heard a loud voice; and the heavens were veiled; and all the creations of God mourned; and the earth groaned; and the rocks were rent; and the saints arose, and were crowned at the right hand of the Son of Man, with crowns of glory.
Here it speaks of the time of the crucifixion–verse 55, and all the creations of God mourned. Why would all the creations of God mourn over the death of Jehovah if each world had its own Savior. This same concept is voiced in greater detail in Doctrine and Covenants 38:11-12–
For all flesh is corrupted before me; and the powers of darkness prevail upon the earth, among the children of men, in the presence of all the hosts of heaven—
Which causeth silence to reign, and all eternity is pained
Clearly, from this scripture we understand that the other worlds know about this earth because their Savior accomplished here the great atoning sacrifice that paid for their sins if they would but repent. And what they witnessed was so horrible that it left them speechless. And all eternity is pained because all eternity was involved!
And now, since it is much too long to print in this column, I will refer you to Doctrine and Covenants 88:42-61 which clearly explains that the Savior visits all his worlds. And thus they all received the light of the countenance of their lord, every man in his hour, and in his time, and in his season–. That would never happen if there were a Savior for each world.
Now you state that since there is an Adam and Eve for each world there must be a Satan for each world. You may be a little confused with your comparisons. It would be appropriate to put Satan in juxtaposition with the Savior rather that with Adam and Eve. It is obviously essential that each earth must have its first progenitors, but not necessarily its own Satan. Satan is not an office. Satan is an individual. One who fell from glory (Lucifer) and became the arch enemy of God.
In Doctrine and Covenants 29:36-37 we read the following–
And it came to pass that Adam, being tempted of the devil–for, behold, the devil was before Adam, for he rebelled against me, saying, Give me thine honor, which is my power; and also a third part of the hosts of heaven turned he away from me because of their agency;
And they were thrust down, and thus came the devil and his angels.
The rebellion of Satan was against Jehovah, who is speaking in this passage– he rebelled against me–not against Adam. It would not be difficult to conclude that Satan opposes Jehovah on all the worlds inhabited by our Father’s children. Certainly there are a third part of the hosts of heaven under his command.