The essence of the question seems to be about whether Morgoth will ever return, and the nature of Sauron, his chief servant.
By the time of The Lord of the Rings, the depredations of Morgoth are largely forgotten, except by the elves and other immortals. Sauron is normally "the Enemy" that the Free Peoples refer to; on the rare occasions that Melkor is mentioned, he is usually referred to as "the Great Enemy," "of which Sauron was only a servant." The sole exception is Treebeard, who is older than the sun and who simply refers to Melkor as "the Enemy." However, the elves are aware that any fight against Sauron is, by proxy, a fight against the power of Morgoth—the "black enemy" of the world, as Feanor named him—because all corruption originated with Melkor.
Sauron was the greatest of the Maiar who were corrupted by Melkor; the lesser corrupted Maiar became the balrogs, most of whom were destroyed in the First Age. Sauron was not evil originally (as Gandalf, who is also a Maia, and was thus probably personally acquainted with Mairon, as he was originally known in Arda), but he was drawn to Melkor's desire for an absolute monarchic order. The Ainur, like the elves and men, were gifted with free will—in particular, the freedom to choose good or evil while they remained alive in Arda. Sauron, although he was older than the world, was permanently corrupted by Melkor's evil. (For an utterly absolute ruler such as Melkor, there was probably no meaningful distinction between servant and lieutenant.) In fact, Morgoth's corruption extended to Sauron and the balrogs, but also far beyond that. From the point in the Spring of Arda when the poisons began to seep out of Utumno while the Valar slept, the world had been essentially permanently polluted by Melkor's maleficence (becoming "Arda marred"). Everything imperfect in Arda before the awakening of the elves was due to Melkor's dark influence.
The Valar were able to, firstly, bind Morgoth, then after his repentance was shown to be false (leading to the Darkening of Valinor, with the destruction of the two trees, followed by the War of the Jewels), then secondly, exile him to the outer darkness beyond the world (after the War of Wrath). However, the Valar did not have the power to send him back to the Timeless Halls, where they dwelt before the creation of the world, so there was always the danger of his returning to pursue more evil.
The nature of Morgoth's final return is unclear. Tolkien's earlier vision of it was expressed in the Second Prophecy of Mandos, which appeared at the end early drafts of the Silmarillion:
When the world is much older, and the Gods weary, Morgoth will come back through the Door, and the last battle of all will be fought. Fionwë will fight Morgoth on the plain of Valinor, and the spirit of Túrin shall be beside him; it shall be Túrin who with his black sword will slay Morgoth, and thus the children of Húrin shall be avenged. In those days the Silmarils shall be recovered from sea and earth and air, and Maidros shall break them and Belaurin with their fire rekindle the Two Trees, and the great light shall come forth again, and the Mountains of Valinor shall be levelled so that it goes out over the world, and Gods and Elves and Men shall grow young again, and all their dead awake.
However, Christopher Tolkien did not include this in the published Silmarillion, for a number of reasons. His father's views on many of the factors involved had evolved over time, although they were not set down in a definitive form. The apparent immortality of Túrin was an issue, as was the question of which of the Children of Ilúvatar would participate in the Second Music, which would remake the world anew. There was the also the fact that Tolkien, in later years, apparently gave up the conceit (or at least considered giving up) that Middle-earth was really the ancient history of the real planet Earth.
However, we can still get a rough idea of what will happen at the End of Days, as Tolkien envisioned it: When the guard upon the Door of Night grows sleepy, Morgoth will return. He will darken the light of the sun and moon, until Eärendil (visible to us as a dawnstar Venus, by the light of the Silmaril he carries) forces the Dark Lord to earth. The forces of evil will flock to his banner, and the greatest war of all time will ensue. The details of Morgoth's death are unclear, since the original conception—which Tolkien was apparently later unhappy with—was that he would be slain by Túrin. However, he will eventually be defeated, and in the process, the world will be so damaged and torn up that the Silmarils lost in the depths of the earth and the sea will be accessible again. With the third Silmaril borne by Eärendil in the sky, all the true light of the Two Trees will be accessible again. Fëanor will finally repent his crimes and be released from the Halls of Mandos. He will bear the Silmarils to Yavanna and reveal the secret of how they were made, so that they may be broken open and the true light of Telperion and Laurelin will be revealed. With this light, the trees may be recreated, as part of the Second Music, and the universe will be remade, lighted once again by the trees' brilliance.