As for the question whether Men could learn "magic", for sure. We are told about "sorcery" in the hands of Black Númenoreans, the Mouth of Sauron is said to be a great sorcerer.
The problem is, Men do not seem to have the, let's say, capacity for magic. Usually when they use it, it is something twisted and evil. The sorcerers are all the servants of Sauron or Morgoth; the description of Gondor in the days of its decay, as much as Númenor in its decay, hint at astrologists and alchemists and so on who had tried to preserve their lives through magic.
So simply put, "magic" is mostly reserved for the non-human races. Elves, of course, have their own "magic", Dwarves also seem to have some (which seems to rest mostly in enchanting weapons or toys for the children of Dale, although there is this one rather cryptic reference of Thorin's during his talk with Bilbo in Bag End, concerning their vengeance against Smaug that has something to do with curses - not sure what this is supposed to mean, or whether it's only an idiom). Istari use "magic" because they are Maiar, after all, so they are mostly using their innate power (with the possible exception of late Saruman, who is said to had been studying "the arts of the Enemy").
One more note to the word "magic" - and the reason I am using the quotation marks - it would seem that Tolkien tried very hard not to use that word, or to make it clear that magic is a misinterpretation of humans, the word for something that does not exist. It seems so that there was (generally, mostly) only "craft" or "art" (of the Dwarves of Elves), the "power" coming from one's special status (of the Valar or Maiar), and then "sorcery" coming from some evil source (all the Ringwraith and "sorcerers" serving Morgoth/Sauron). "Magic" is a human word that mixes all those up, because humans can't understand these concepts, just like Galadriel had told Sam:
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For this is what your folk would call magic, I believe, though I do not understand clearly what they mean, and they seem also to use the same word of the deceits of the Enemy.
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So as for a "random" wizard figure existing in Middle-Earth, no. At least not a "good" wizard - some evil sorcerer, yes. Merlin however is a special case, because he is, after all, not supposed to be purely human, or at least he seems to be gifted by some supernatural power. So Merlin
specifically could possibly exist in Middle-Earth, but any other random wizard (as in, Merlin taking a random boy for an apprentice), nope.