Was Evil Eradicated From The World When Sauron Was Destroyed?
Certainly not. Evil still exists, but it won't be represented by a single entity in the future, until, perhaps, the end of the world.
"After which the Third Age began, a Twilight Age, a Medium Aevum, the first of the broken and changed world; the last of the lingering dominion of visible fully incarnate Elves, and the last also in which Evil assumes a single dominant incarnate shape."
-Tolkien, Letter #131
"never again (unless it be before the great End) will an evil daemon be incarnate as a physical enemy; he will direct Men and all the complications of half-evils, and defective goods, and the twilights of doubts as to sides, such situations as he most loves (you can see them already arising in the War of the Ring, which is by no means so clear cut an issue as some critics have averred): those will be and are our more difficult fate."
-Tolkien, Letter #156
Other evils there are that may come; for Sauron is himself but a servant or emissary... Yet it is not our part to master all the tides of the world, but to do what is in us for the succour of those years wherein we are set, uprooting the evil in the fields that we know, so that those who live after may have clean earth to till. What weather they shall have is not ours to rule."
-Gandalf, The Return of the King, Book V, Chapter 9: "The Last Debate"
"For though Sauron had passed, the hatreds and evils that he bred had not died, and the King of the West had many enemies to subdue before the White Tree could grow in peace. And wherever King Elessar went with war King Éomer went with him; and beyond the Sea of Rhûn and on the far fields of the South the thunder of the cavalry of the Mark was heard, and the White Horse upon Green flew in many winds until Éomer grew old."
-The Lord of the Rings, Appendix A: "Of the House of Eorl"
Yet the lies that Melkor, the mighty and accursed, Morgoth Bauglir, the Power of Terror and of Hate, sowed in the hearts of Elves and Men are a seed that does not die and cannot be destroyed; and ever and anon it sprouts anew, and will bear dark fruit even unto the latest days.
-The Silmarillion, "Of the Voyage of Earendil and the War of Wrath"
As Tolkien put it, the destruction of the Ring was not the end of evil, but merely:
The end of visibly incarnate Evil.
-Tolkien, Letter #191
In the future, then, no one entity will be as evil as Sauron, but there will be a far greater number of lesser evil people and things. This will make each evil easier to defeat, but will also make it all but impossible to conquer evil altogether.
And this is what we see in our own time - perhaps the most evil person of the past century was a laughable buffoon, whom Tolkien described as:
"that ruddy little ignoramus, Adolf Hitler"
-Tolkien, Letter #45
Hitler's regime, as monstrous and appalling as it was, still gave rise to the phrase "the banality of evil" ("banal" is defined as "so lacking in originality as to be obvious and boring"). Whatever Sauron may or may not have been, he was certainly not "banal".
The evils we face are less formidable but more numerous, less singular but more common, less incarnate but more insidious. Tolkien knew a great deal about such things, having lived through the horrors of the World Wars. Whereas the Men of the West were able, for the most part, to unite in the face of the singular physical evil presence called Sauron, in the future, evils would be at all sides, and we would fracture and squabble amongst ourselves rather than uniting to confront them.
Tolkien commented on the war effort, in the closing months of WWII, in a letter dated 1944:
An ultimately evil job. For we are attempting to conquer Sauron with the Ring. And we shall (it seems) succeed. But the penalty is, as you will know, to breed new Saurons, and slowly turn Men and Elves into Orcs. Not that in real life things are as clear cut as in a story, and we started out with a great many Orcs on our side.
-Tolkien, Letter #66.
Clearly, in Tolkien's view, evil survived, and thrived, after Sauron's fall. We are still dealing with the repercussions today.
But we must not despair; there will always be evil, but there will also always be love, and hope, and good; and fortunately for us, good, love, and hope will always be more powerful than evil, hatred, and despair. We might say, as Tolkien's characters do:
“The world is indeed full of peril, and in it there are many dark places; but still there is much that is fair, and though in all lands love is now mingled with grief, it grows perhaps the greater.”
-Haldir, The Fellowship of the Ring, Book II, Chapter 6: "Lothlòrien"
Did Tolkien Ever Write About Middle-Earth After The War Of The Ring?
Yes, a bit. Three drafts and one fragment were produced for a story titled The New Shadow.
"I did begin a story placed about 100 years after the downfall of Sauron, but it proved both sinister and depressing. Since we are dealing with Men it is inevitable that we should be concerned with the most regrettable feature of their nature: their quick satiety with good. So that the people of Gondor in times of peace and justice and prosperity would become discontented and restless - while the dynasts descended from Aragorn would become just kings and governors - like Denethor or worse. I found that even so there was an outcrop of revolutionary plots, about a centre of secret Satanistic religion; while Gondorian boys were playing at being Orcs and going around doing damage. I could have written a 'thriller' about the plot and its discovery and overthrow - but it would be just that. Not worth doing."
-Tolkien, Letter #256
But Tolkien's mind was not made up, despite the indications in the above quote that he had permanently given up on the story. The letter quoted above was written in 1964, but in 1968, Tolkien again began working on The New Shadow. He revised the existing passages and the, for reasons which remain unclear, abandoned the story at the same point where his work had stopped years earlier. Then, in 1972, he wrote to a friend:
I have written nothing beyond the first few years of the Fourth Age. (Except the beginning of a tale supposed to refer to the end of the reign of Eldarion about 100 years after the death of Aragorn. Then I of course discovered that the King's Peace would contain no tales worth recounting; and his wars would have little interest after the overthrow of Sauron; but that almost certainly a restlessness would appear about then, owing to the (it seems) inevitable boredom of Men with the good: there would be secret societies practising dark cults, and 'orc-cults' among adolescents.)
-Tolkien, Letter #338.
The fragmentary The New Shadow manuscript, only 13 pages long, was later published by Christopher Tolkien in The Peoples of Middle-earth, in the multi-volume series The History of Middle-earth. The version that appears in this volume is actually composed of different parts of 3 separate manuscripts (and a few lines scribbled on an envelope dated to 1968), written years apart from one another.
There are significant problems with the dates in the manuscripts. Despite what he says in letter 338, the story itself says that the events being described begin 105 years after the destruction of Sauron. This is very peculiar, because the story also says that Aragorn is dead and his son Eldarion is now king; however, the appendices to The Return of the King make it clear that Aragorn died more than 120 years after Sauron was destroyed, and only then did his son succeed him. So unless he later corrected this mistake, a serious inconsistency would have occurred.
Ironically, the second paragraph of The New Shadow addresses the first point in your question:
"'Deep indeed run the roots of Evil,' said Borlas, 'and the black sap is strong in them. That tree will never be slain. Let men hew it as often as they may, it will thrust up its shoots again as soon as they turn aside. Not even at the Feast of Felling should the axe be hung upon the wall!'"
-The History of Middle-earth, Volume XII: The Peoples of Middle-earth, Chapter 16: "The New Shadow"
[Note: Borlas is Beregond's younger son, and Bergil's younger brother; Beregond was the Guard of the Tower who befriended Pippin and, saved Faramir's life, and later became the head of Faramir's guard; Bergil's was Beregond's older son, and also befriended Pippin]
A PDF file of the entire text of The Peoples of Middle-earth, including the unfinished manuscript of The New Shadow, is available online, and very easy to find, but I won't post a link here because it probably violates all kinds of copyright laws. You should be able to find it without any difficulty, if you are so inclined.
However, I must warn you that if you read The New Shadow, you will find that the incredibly brief story cuts out just as it is getting interesting, and no Tolkien fan can read it without being tormented by thoughts of what might have been. It is tantalizing, but so short that it whets the appetite without any possibility of satisfying the hunger it provokes. It is almost cruel in its brevity.
More generally, the appendices at the end of The Return of the King contain snippets of information about what happened immediately after the events of The Lord of the Rings: Eomer growing old, Aragorn and Arwen dying, etc. Tolkien didn't consider the events that took place after the War of the Ring to be worth publishing, as his letters regardless The New Shadow suggest. However, Tolkien also said that the Ring was destroyed 6,000 years ago in our own world, so everything you learned about (dating from after that period)in history class can be considered to have followed the War of the Ring.
Note: The passage I quoted from Letter #45, in highly abbreviated form, is worth quoting at greater length:
"Anyway, I have in this War a burning private grudge—which would probably make me a better soldier at 49 than I was at 22: against that ruddy little ignoramus Adolf Hitler (for the odd thing about demonic inspiration and impetus is that it in no way enhances the purely intellectual stature: it chiefly affects the mere will). Ruining, perverting, misapplying, and making for ever accursed, that noble northern spirit, a supreme contribution to Europe, which I have ever loved, and tried to present in its true light."
―J.R.R Tolkien, Letter #45