I have a couple questions I have been pondering and haven’t been able to find an answer.
Was Satan part of Heavenly Father’s original plan?
Is Satan the prodigal son?
Does Satan have power in other worlds or just ours?
Let me respond to your questions in the order you asked them. Our scriptures tell us that the Father presented his plan, and then Lucifer rebelled against it. (See Moses 4:1,Abraham 3:24-28, and D&C 29:36-38). As part of His plan, the Father solicited a Savior. The Firstborn agreed to fill this duty as the Father had said. Lucifer said he would take the role, but under different conditions, and should thus receive the Father’s honor. So Satan’s rebellion appears to have been an aberration of the Father’s plan, not an intended part of it.
It is worth noting that the word “Satan” means “adversary.” As father Lehi made clear, adversity is an integral part of the plan of happiness. So there had to be opposition to bring about the Father’s plan of salvation. But it would be a mistake to think that the Father appointed Satan or his evil followers to the course they chose. The Father seamlessly wove Satan’s rebellion into the working of the plan of salvation, so that it would go forward regardless of Satan’s evil actions. But the Father did not ordain Satan to rebel. God is not the father of evil.
The parable of the prodigal son is the last of three parables that the mortal Savior used to show the value of finding the lost human soul. When we go astray, we do not lose our intrinsic value as long as we turn to the Savior. This cannot apply to Satan or his followers, who have rebelled against their only source of salvation and are thus lost forever. Jesus was speaking of the lost sheep of Israel, not of Satan, when He told this parable.
So far as I understand scripture, Satan and his minions were cast out of our premortal heaven to this world, and are confined to it. Here are some scriptures and other teachings to consider:
Revelation 12:9 And the great dragon was cast out, that old serpent, called the Devil, and Satan, which deceiveth the whole world: he was cast out into the earth, and his angels were cast out with him.
[W]hen the revolt took place in heaven, when Lucifer, the Son of the Morning, was cast out[, w]here did he go? He came here, and one-third part of the spirits in heaven came with him. Do you suppose that one-third part of all the beings that existed in eternity came with him? No, but one third part of the spirits that were begotten and organized and brought forth to become tenants of fleshly bodies to dwell upon this earth. (Journal of Discourses, 3:369)
We have baptized a great many into this Church and kingdom—not many, certainly, when compared to the twelve hundred million inhabitants of the earth—but a great many have apostatized. What! Latter-day Saints apostatize? Yes. I tell you people will apostatize who have received the holy priesthood and Gospel of Jesus Christ, if they do not honor God, if they do not keep his commandments, obey his laws and humble themselves before the Lord; they are in danger every day of their lives. Look at the number of devils we have, round about us! We have I should say, one hundred to every man, woman and child. One third part of the heavenly host was cast down to the earth with Lucifer, son of the morning, to war against us—which I suppose will number one hundred million devils—and they labor to overthrow all the Saints and the kingdom of God. (Journal of Discourses 21:126)
“The punishment of Satan and the third of the host of heaven who followed him, was that they were denied the privilege of being born into this world and receiving mortal bodies. They did not keep their first estate and were denied the opportunity of eternal progression. The Lord cast them out into the earth, where they became the tempters of mankind—the devil and his angels” (Doctrines of Salvation,1:65).