That’s a question that I’m still asking and answering myself!
Proper nouns—such as names of countries, nationalities, and languages—should be capitalized. However, when I’m talking about greyfolk, it’s like talking about humans/humanity, which are concepts that aren’t capitalized. So, ‘the English language’ is capitalized, but the idea of ‘human language’ isn’t.
Of course, I could also call it ‘the Greyfolk language’ if I wanted to, but it’s not the only language of greyfolk. Though, I admittedly get tripped up because I often say ‘the greyfolk’ (species, language, or otherwise) because ‘greyfolk‘ is a collective noun, which means it’s always plural—like ‘people’!—and calling it ‘the greyfolk language’ makes me want to capitalize the ‘g’. Though, to be fair, people’s language makes more sense than just people language. It’s confusing—the term ‘greyfolk’ is used like a common name.
Furthermore, just because I use ‘the’ doesn’t mean I need to capitalize greyfolk if I talk about it in a different way. It’s ‘the language of the people’ or ‘the people’s language’. So, if I’m referring to them as a people or as a species or whatever, then it would be fair to call my language the language of the greyfolk or the greyfolk’s language.
I haven’t answered the question, though; I’ve just been musing and rambling about it. My answer is that I’m tossing the capital ‘G’ because I think it’s a bad habit. I prefer greyfolk (language), (this) greyfolk language, the greyfolk’s language, or (the/this/a) language of the greyfolk. Eventually, it will get a name within the language itself, and then that’s probably what I’ll start using over anything else because it’ll be less confusing.
In the same vein, for the group, I prefer (the) greyfolk (species). So, just ‘greyfolk’ or ‘the greyfolk’ is fine because it should (as long as I remember) imply the species and not the language unless just ‘greyfolk’ is used in a context concerning language. I think Homo cinerium might work too.