Ok, I said in this post that I wanted to talk about POV and what different characters would use for their species; I've been thinking about it for a while, because it's important to me to portray Elves as thinking of themselves as the default and I like having them think of mortals as different, as a way of worldbuilding and establishing POV. It's the same basic principle as why I think the War of the Elves and Sauron is a Númenórean name, not an Elvish one.
In brief: man and woman can and should be used to refer to Elves when writing from an elven POV.
I think about markedness a lot when I think about writing generally, and one of the concepts is that there are things that are unmarked, the standard, the default, the normal, the unexceptional. To an elf, the default species is elf, not human or dwarf or ent. I want to write from that POV - to an elf, humans are different, even alien. They're other. Elves are us, humans are them.
I, meanwhile, am writing in English, and man and woman are common words, the most "unmarked" words for the concepts; they're standard (e.g., woman sounds more normal than female human). Something like elf woman or female elf is more marked. Why would an elf use a more marked term to refer to their own race? Using she-elf or whatever for Elves but woman for humans uses more marked words (more uncommon, more specific, etc) for elves than it does for humans - but for elves, it should be the opposite! Elves are unmarked! Elves are default! Humans are different. If an elf is talking about someone, the default assumption would be that the someone is an elf, so if they're not an elf, it would be specified.
So when writing from the POV of an elven character, I would use woman for a female elf and mortal woman for a female human (if necessary to disambiguate). Man, meanwhile, means "Human" when it's capitalized and male when it's not. (Now, if I were writing from a human or dwarven pov, I would use elven woman, if necessary to specify that she's an elf.)
But, you say, what about using Elvish words? Well, first of all, I hate it. We're not writing in Quenya. English is great, and so are the other human languages people use to write fanfic. But that is a subjective matter of taste and you may disagree! Nothing wrong with that, de gustibus, etc.
More objectively, nér and nis are not words specific to Elves; nér means all males, so using it for specifically elf men and not human and dwarven men is incorrect (to quote Elfdict, "Nér can be used regardless of species and so is equally applicable to male Elves, Men, or Dwarves, but is unlikely to be used of male animals, for which the word [ᴹQ.] hanu is more applicable."") (Sindarin is a little more complicated, given the more complex out-of-universe changes, but it too has race-neutral terms for man and woman.)
Lastly, Tolkien himself uses man and woman to refer to non-human species. He calls Galadriel and Finduilas women, and Aredhel is "taller than a woman's wont." Earendil is a man (though he is not a Man) and Curufin is a horseman and there are lots and lots of kinsmen and kinswomen. Hobbits meanwhile, in Appendix F, have "women" and "man-children" (Tolkien is talking about how Hobbits name babies, thus the children part, but I like to think he's getting one more dig in at Pippin).
Sometimes I'm tempted to go on a big long rant about what I think are good and bad ways to portray multilinguality in fiction and always end up refraining because it seems a little too arrogant. Hi, undercat, you actually just covered a big chunk of it!
Why would an elf use a more marked term to refer to their own race? Using she-elf or whatever for Elves but woman for humans uses more marked words (more uncommon, more specific, etc) for elves than it does for humans - but for elves, it should be the opposite! Elves are unmarked! Elves are default! Humans are different. If an elf is talking about someone, the default assumption would be that the someone is an elf, so if they're not an elf, it would be specified.
This bit is so important because it gets at the heart of what bugs me about so much handling of conlangs in fiction. Your point of view character is supposed to be your readers' lens into the world, the eyes and mind through which the story happens. If general you take a concept that is perfectly ordinary for your POV character, and for no good reason you turn it strange and exotic-seeming through vocabulary like nér or elf-man when there are serviceable regular English words right there, what you are doing at this point is undermining your POV. It's like you-the-author is taking a break from writing the story just to go "oh btw these are ELVES did I mention look isn't it strange they're ELVES not like you or me", a little like those in-line author's notes that used to be around which I have thankfully not seen for a long time because egads, they were terrible. It's alienating the readers from the character and inaccurately representing their POV when that is the precise opposite of what you want to do in a third person limited fic.
Especially because, like... you can actually do a ton of subtle character work involving what groupings your POV character uses. Maybe one character distinguishes man versus mortal man versus dwarf (gender unspecified). A Sinda from Doriath might have fellow Iathrim as their unmarked group and split up other Elves into Golodhrim/Mithrim/Falathrim/Laegrim (using, one notes, the impolite word for Noldor and ignoring Avari entirely). A Noldo might split the Noldor into various groups depending on liege lords but chuck the Sindar into one whole. Someone familiar with humans might split up mortal man depending on which of the Houses they're from, whether they're Édain, an Easterling or other, etc. Someone familiar with Dwarves might split them up depending on what city they're from, Longbeard vs Broadbeam vs Firebeard etc... and maybe also know how dwarven gender presentation works. Hell, maybe some teenager from Sirion doesn't actually distinguish between mortals and Elves most of the time but uses refugee place of origin. There is seriously no end of potential for interesting characterisation opportunities here. But before you can do anything here, you first have to get out of your own way and figure out how your character would describe the people around him.
