Three Rings for the Elven-kings under the sky,
Seven for the Dwarf-lords in their halls of stone,
Nine for Mortal Men doomed to die,
In the excerpt above, is there any significance to the bit about "Mortal Men"? Obviously men of Middle-earth are mortal, but aren't Dwarves mortal as well?
The Doom of Man, the mortality of humans, is a recurring theme in Tolkien's writing.
The Elves, of course, are immortal. They don't die in flesh from natural causes, only from violence (or heartbreak). But even when they do, their spirits go to the Halls of Mandos, where they spend some time but sometimes come back again, if not immediately, then after the end of the world. For Tolkien, immortality isn't physical immortality, but a return of the spirit into the grand scheme of things, again and again.
For the Elves die not till the world dies, unless they are slain or waste in grief [...] and dying they are gathered to the halls of Mandos in Valinor, whence they may in time return.
(Quenta Silmarillion, Chapter 1: Of The Beginning of Days)
So, to establish the basic concepts we'll use, immortality, in Tolkien's works, means being tied to the world, to the fate of Arda. Not dying in flesh is a part of it, but the essence is having your spirit wait until needed again.
Of the Fate of the Dwarves
The Dwarves are a bit of a complicated issue. They were made by Aulë against the wishes of Ilúvtar, and he is responsible for their fate:
Aforetime it was held among the Elves in Middle-earth that dying the Dwarves returned to the earth and the stone of which they were made; yet that is not their own belief. For they say that Aulë the Maker, whom they call Mahal, cares for them, and gathers them to Mandos in halls set apart; and that he declared to their Fathers of old that Ilúvatar will hallow them and give them a place among the Children in the End. Then their part shall be to serve Aulë and to aid him in the remaking of Arda after the Last Battle.
(Quenta Silmarillion, Chapter 2: Of Yavanna and Aulë)
So regarding your second question, the dwarves are also immortal, in the same sense as the Elves: even if they die, their spirit lives on. They share the same ties to the world of Arda that the elves do.
Of the Doom of Man
But humans are a different matter:
What may befall their spirits after death the Elves know not. Some say that they too go to the halls of Mandos; but their place of waiting there is not that of the Elves, and Mandos under Ilúvatar alone save Manwë knows whither they go after the time of recollection in those silent halls beside the Outer Sea. None have ever come back from the mansions of the dead, save only Beren son of Barahir, whose hand had touched a Silmaril; but he never spoke afterward to mortal Men. The fate of Men after death, maybe, is not in the hands of the Valar, nor was all foretold in the Music of the Ainur.
(Quenta Silmarillion, Chapter 12: Of Men)
So men, unlike elves and dwarves, do die - they die entirely, and go away somewhere never to return. Some elves have expressed surprise, pity or even jealousy at this fate, but it's something that Tolkien refers to again and again, how with their short lifespan and unknown fate, Men are driven to do more, and thus achieve more despite their drawbacks (emphases mine):
It is one with this gift of freedom that the children of Men dwell only a short space in the world alive, and are not bound to it, and depart soon whither the Elves know not. Whereas the Elves remain until the end of days, and their love of the Earth and all the world is more single and more poignant therefore, and as the years lengthen ever more sorrowful.
(Quenta Silmarillion, Chapter 1: Of The Beginning of Days)
It's not entirely certain that men's spirits die forever when they die. It's actually hinted that they do have an afterlife somewhere. But the scope of Tolkien's writing is the fate of Arda and Middle Earth, and the Valar only have knowledge of that world. And the fate of Men is determined by Ilúvatar and taken outside the borders of Arda, so that as far as Middle Earth is concerned, once dead, they're gone, free from worrying about the world's fate - that is why some elves see it as freedom.
