When Frodo Arrived In The Undying Lands If He Found Out About The Halls Of Mandos And Decided To Spend The Rest Of His Days Sitting Peacefully In Mandos Halls Would Manwe And Mandos Have Allowed Him? If yes Why? If Not Why?

Sam's going over the Sea is a tradition handed down from his daughter Elanor, in her family of the Fairbairns of the Towers, so a bit conjectural.So Frodo went not to Valinor (Aman), but to Tol Eressëa within sight of Eldamar (the coastal region of Valinor inhabited by the Eldar), where he lived and died. The same is true of Bilbo, of Sam, and of Gimli, all of whom were permitted to sail to Eressëa in the Uttermost West, where after a time, they all died.
The alliteration is strong with this one...(fodder for a fanfic?)
My guess is that it was Rose who kept Sam grounded. Sam suffered from his passage through Mordor as did Frodo, but not so intensely; and it was only after bearing the Ring for a short while himself that Sam finally had pity for Gollum and – fortunately for Sam and everyone else – showed him mercy. I imagine that Tolkien’s abandoned Epilogue (which he described as “universally condemned” in Letter 144) would, if properly completed, have displayed this: at the end of one version published in Sauron Defeated, Sam nearly breaks down, but Rose keeps him stable. Once she died, he no longer had that stabilization, and the only good option open to him was to sail into the West, an outcome first foreshadowed at the beginning of Fellowship of the Ring in “Shadow of the Past”, when Sam “half chant[ed],” “They are sailing, sailing, sailing over the Sea, they are going into the West and leaving us,” and again in “Scouring of the Shire” in Return of the King when Ted Sandyman sneers, “I thought you’d gone off in one o’ them ships you used to prattle about, sailing, sailing. What d’you want to come back for?”
Ah, Perry Mason.
I always assumed that even though Frodo received some healing during his stay on Tol Eressea, he was so damaged by his experience, he didn't live for more than a few years there.
Here I think the answer would be a definite yes. The Istari were (severely!) limited as long as they were in Middle-earth (including the returned Gandalf, in his incarnation as The White, if less so than as The Grey). If the Elves were living libraries (like Elrond and Galadriel), how much more so would a no longer fettered Maia be? And Olórin had a special liking of the Elves in Valinor, and of the Hobbits in Middle-earth. Add to that Frodo's experience during the quest, which set him apart from all Little and Big Folk (and in the latter I would include even Aragorn!) - yes, they would still know each other, I would think.
Reading that sent a shiver down my spine.Frodo didn’t go to Valinor. He went to the island of Tol Eressëa in the Bay of Eldamar. It was forbidden for mortals to step ashore the coasts of Valinor. (This was in fact one of the charges Irmo Mandos made against Eärendil.)
In a footnote to Letter 297, Tolkien wrote,
Frodo … as a special grace … [was] granted a purgatorial (but not penal) sojourn in Eressea, the Solitary Isle in sight of AmanAnd in Letter 246,
Frodo was sent or allowed to pass over Sea to heal him – if that could be done, before he died. He would have eventually to “pass away”: no mortal could, or can, abide for ever on earth, or within Time. So he went both to a purgatory and to a reward, for a while: a period of reflection and peace and a gaining of a truer understanding of his position in littleness and in greatness, spent still in Time amid the natural beauty of “Arda Unmarred”, the Earth unspoiled by evil.Bilbo went too. … His companionship was ... necessary for Frodo's sake – it is difficult to imagine a hobbit, even one who had been through Frodo's experiences, being really happy even in an earthly paradise without a companion of his own kind… But he also needed and deserved the favor on his own account. He bore still the mark of the Ring that needed to be finally erased: a trace of pride and personal possessiveness.So Frodo went not to Valinor (Aman), but to Tol Eressëa within sight of Eldamar (the coastal region of Valinor inhabited by the Eldar), where he lived and died. The same is true of Bilbo, of Sam, and of Gimli, all of whom were permitted to sail to Eressëa in the Uttermost West, where after a time, they all died.