The reasons that Frodo's wounds cannot be fully healed upon Middle-earth are complex. To begin with, he is wounded by the morgul blade, which in itself is not something easily recovered from. But then he had taken up the quest and given all of himself to reach Mount Doom, stretching his limits physically as well as mentally by refusing to give in to the ring. But at last Frodo reaches the very place of the One Ring's forging, where of course its temptation is greatest: "At the last moment the pressure of the Ring would reach its mazimimm - impossible, I should have said, for any one to resist, certainly after long possession, months of increasing torment, and when starved and exhausted" (Letter 246).
Thus, Frodo is not responsible for taking up the Ring at Mt. Doom rather than throwing it to its destruction. He failed his quest, but the fact that there was no possibility of fulfilling it shows it wasn't a moral failure. Frodo's efforts, humility, and pity, have brought the Ring to a place where its destruction is possible, and indeed the ring is destroyed.
But Frodo has not given up the Ring freely, but rather it has been taken from him by force (which as Gandalf said would break his mind, although this is Gollum taking it not Gandalf). Consequently, Frodo almost wishes the Ring was not destroyed; awakening from sickness, Frodo says "it is gone for ever, and now all is dark and empty" ("The Grey Havens"). The Ring's abscence has left an impossibility for all recovery from sickness, so indeed his action upon Mt. Doom, although inevitable, makes it necessary for him to leave Middle-earth if he wants to heal.
As for the question about mortals in the Undying Lands, it is not the Land that makes the people there immortal, but the nature of the people. Mortals cannot become Immortal (death is an inherent quality, biological like in a sense, of the mortal races) except through an act of Eru, which only occurs for Tuor and the half-elven. The visit to Aman is "strictly a temporary reward: healing and redress of suffering. They cannot abide for ever, and though they cannot return to mortal earth, they can and will 'die' - of free will, and leave the world" (Letter 154).
Don't take this death at free will idea to mean that they can live for ever, because it is only an idea that those who accept their mortality actually realize. The lifespan of a mortal "cannot really be increased qualitatively or quantitatively; so that prolongation in time is like stretching a wire out ever tauter, or 'spreading butter ever thinner' - it becomes an intoleable torment" (Letter 131). (the immortality of a ringwraith is not to be envied).
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"He was sent by a mere prudent plan of the angelic Valar or governors; but Authority had taken up this plan and enlarged it, at the moment of its failure."
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