Is the word "зелёный" a noun here?: Бриллиантовый зелёный
I think that in this case the entire phrase is a noun form, it's not that зеленый is a noun, and бриллиантовый is an adjective; otherwise you should have been able to attach other adjectives to зеленый to refer to other kinds of зеленый, yet there's only one unique entity called бриллиантовый зеленый.
It is called субстантивированное прилагательное, "substantivized adjective", an adjective used as a noun. There are a lot of them in Russian, and usually they have a generic noun associated with them, but it is usually elided. As for
бриллиантовый зелёный the generic noun is "краситель" (pigment, dye), not "color" as @fedja suggested.
Almost, but not quite. I'm not sure how the corresponding grammar construction is called (it is some kind of ellipsis), but what is going on is that there is an omitted noun ("color"), which doesn't appear because the speaker wants to be brief but which transfers all its grammatical properties to the adjective "green". Compare with
— Какая машина твоя?
— С прицепом.
— Там две таких. Которая с прицепом?
— Синяя с прицепом, без правого зеркала.
— Да у твоей с прицепом все шины спущены!
Note that if the first question were
— Какой автомобиль твой?
the same combination of words "с прицепом" would borrow the masculine gender and the following dialog would read
— С прицепом.
— Там два таких, который с прицепом?
— Синий с прицепом, без правого зеркала.
— Да у твоего с прицепом все шины спущены!
so it is clear that while the role of с прицепом in the sentence is pretty much indistinguishable from the role of a noun, the true noun ("car") is hiding behind it and the best name for с прицепом would be something like a "noun impostor" (there is no such term, of course, but that's more or less what it really is).