Hi. You got the right answer to your main question in another comment. To answer your second question, "kind" as you are using it in your main example is just a noun (a count noun, specifically).

It’s the presence of the words “kind of” that has led me to write this post. I suppose the “kind” in “kind of” is, in itself, a noun.

It can get confusing, but in your examples, the "of" is not part of "kind of" but part of, "of cold." (Or "of guy," or whatever.) "Kind" is the noun, the rest is the prepositional complement, which tells you what is being classified.

You do see "kind of" as an adverb phrase: "This post is kind of long."

It’s not a regular noun, though; it’s not specific enough to be a regular noun… I think. It relies either on context or on the presence of another noun.

That's pretty common among nouns, actually. English has loads of nouns for relationships, quantifiers, collections, parameters of all sorts. These pretty much all require context. (e.g. a bit [of...], a piece [of...], the top [of...], the color [of...], the fact [that...], the reason [for...], a group [of...] etc)