Do Mormons believe that the fall was likewise needed on each inhabited planet?
No authoritative LDS doctrine on this subject, except that the first man and woman on other worlds had the name Adam and Eve.
33 And worlds without number have I created; and I also created them for mine own purpose; and by the Son I created them, which is mine Only Begotten.
34 And the first man of all men have I called Adam, which is many.
26 And Adam called his wife’s name Eve, because she was the mother of all living; for thus have I, the Lord God, called the first of all women, which are many.
Do they believe that Jesus also eventually died on behalf of the inhabitants of each planet?
Mormons believe that Jesus only needed to perform the Atonement once because it was an infinite and eternal sacrifice.
Doctrine and Covenants 76:22-24,40-43
22 And now, after the many testimonies which have been given of him, this is the testimony, last of all, which we give of him: That he lives!
23 For we saw him, even on the right hand of God; and we heard the voice bearing record that he is the Only Begotten of the Father—
24 That by him, and through him, and of him, the worlds are and were created, and the inhabitants thereof are begotten sons and daughters unto God.
40 And this is the gospel, the glad tidings, which the voice out of the heavens bore record unto us—
41 That he came into the world, even Jesus, to be crucified for the world, and to bear the sins of the world, and to sanctify the world, and to cleanse it from all unrighteousness;
42 That through him all might be saved whom the Father had put into his power and made by him;
43 Who glorifies the Father, and saves all the works of his hands, except those sons of perdition who deny the Son after the Father has revealed him.
In Times and Seasons 4 no. 6 WW Phelps in a poem verse based on section 76 wrote:
By him, of him, and through him, the worlds were all made,
Even all that career in the heavens so broad.
- Whose inhabitants, too, from the first to the last,
Are sav'd by the very same Saviour (Savior) of ours;
And, of course, are begotten God's daughters and sons,
By the very same truths, and the very same pow'rs.
On why the Atonement happened here and not another world Joseph Fielding Smith speculated: (Signs of the Times(this is just a link to amazon) p 5 or 12)
Perhaps this is the reason Jesus Christ was sent here instead of some other world, for in some other world they would not have crucified Him, and His presence was needed here because of the extreme wickedness of the inhabitants of this earth
Brigham Young in Journal of Discourses seems to agree that this world is one of the more wicked worlds.
I suppose that God never organized an earth and peopled it that was ever reduced to a lower state of darkness, sin and ignorance than this. I suppose this is one of the lowest kingdoms that ever the Lord Almighty created, and on that account is capable of becoming exalted to be one of the highest kingdoms that has ever had an exaltation in all the eternities
See also Alma 34:9-10,12-13
However Brigham Young taught that each world had its own Adam and Eve, and its own savior, however, please note that this is not considered final or authoritative.
Let me open the eyes of your understanding. There has never been a time when the creations of worlds commenced. They are from eternity to eternity in their creations and redemption. After they are organized they experience the good and the evil, the light and the dark, the bitter and the sweet as you and I do. There never was a time when there were not worlds in existence as this world is, and they pass through similar changes in abiding their creation preparatory to exaltation. Worlds have always been in progress, and eternally will be.
Every world has had an Adam and an Eve, named so simply because the first man is always called Adam and the first woman, Eve. And the oldest son has always had the privilege of being ordained, appointed and called to be the heir of the family if he does not rebel against the Father, and he is the Savior of the family. Every world that has been created has been created upon the same principle. They may vary in their varieties, yet the eternity is one: it is one eternal round.
emphasis added by me