“Magic” in a work of fiction has never, for me, been better evoked than in the conclusion to “Lothlórien” and in the following chapter, “The Mirror of Galadriel”. It is not "wizard magic", the bending of the world to the caster’s will, such as Gandalf or Saruman conjuring fire, but "Elf magic", an essence of being. It is explained, as much as it can be explained, in The Mirror of Galadriel, but that wave of strangeness is first identified by Sam at Cerin Amroth, who Frodo saw
...was now standing beside him, looking round with a puzzled expression, and rubbing his eyes as if he was not sure that he was awake.
This is faerie glamour, the magic of the Elves, and this is its heart, but rather than expressing its power, Tolkien has Haldir point out how vulnerable this land is to the threat from Dol Guldur.
"...whereas the light perceives the very heart of darkness, its own secret has not been discovered. Not yet."
Already, Tolkien is hinting that Lothlórien can be but a brief refuge; that the threat is growing and the Fellowship’s travails enjoying but a brief hiatus.
This motif is repeated with Aragorn, who is “revealed” to Frodo by the magic of Cerin Amroth. Aragorn, in whose bloodline runs Man, Elf and Maia,
belongs here, but his doom is at hand. Amidst the sense of wonder and beauty, Tolkien introduces finality, and a possible portent of Aragorn’s demise.
And taking Frodo’s hand in his, he [Aragorn] left the hill of Cerin Amroth, and came there never again as living man.
Edited to fix quote
As for the rapidity that he follows the Fellowship, I will refrain from commenting, as I am notoriously bad with geography (and languages). However, I not sure I agree with your second point. Smeagal's reaction to the Elven rope is precisely because it is made by the Elves. The Mallorn trees are not made by the Elves, even if they only grow in Middle-earth in the heart of Elvendom (of course they also grew in Tol Eressëa and probably Valinor, and in Númenor before its Fall). Like the rest of the kelvar and olvar, they are beings that are created by Ilúvatar, and beloved by Yavanna. As is Sméagol himself.