Indefinite articles as adjectives and adverbs [?]

"Kind" in "a kind of computer" is a noun, not an adjective.

A(n) is not an adverb and, as far as I know, don't have any adverbial usages.

Whether or not a(n) is an adjective, or an article, depends on how you want to define the words "adjective" and "article". A(n) has special behavior and special restrictions that are not shared with (other) adjectives. A(n) also doesn't behave exactly the same as the "definite article" the.

A(n) originated as a reduced form of the numeral one and I've seen proposals in some linguistic literature that it might still be possible to analyze it synchronically as some special version of one (e.g. "The indefinite article – Indefinite? – Article?", Thomas Leu, 2012).

Articles are articles. They cannot be classified as anything else, except in cases where the article in question has other meanings in other contexts (like "A" can be an article, a musical key, and more).

But article is a part of speech in English. There is no need to classify an article as anything else. "A", in particular, is an indefinite article.