Grammar – Use of Determiners as Adjectives Explained

• Not all adjectives have a natural adverbial use. I'd think you'd be hard pressed to use beautiful where beautifully is called for.
• In "She is small," small is an adjective serving as a nominative complement. It's not an adverb.
• It might be best to consider participles as verb forms that have uses as modifiers. When they modify nouns, that's an adjectival use, but it really doesn't make them adjectives. An adjective (say, red) may be compared (to get redder). That won't work for cooked. You'll have to say "more cooked."
• I don't have time to search for a past participle that can't have an adjectival use. Most can.
• In "the book was written", written is part of the past passive; it's not a modifier. You can tell because you can sensibly append a prepositional phrase with by to give who wrote the book. Similarly, "to be cooked" might be passive infinitive if you're concerned about who did the cooking. Same test. If you're concerned with the internal temperature, then it would be a participial modifier.
• Past participles have adverbial (or at least semi-adverbial) uses: "He lived cursed by fate." The participle would say (at least in part) how he lived.

A compound noun is a noun (though it might consist of two orthographic words); some modern dictionaries have adopted the practice of labelling even open compounds 'noun':

school bus n.

A publicly or privately owned vehicle

{AHDEL}

They treat other compounds similarly:

bottle green n. A dark to moderate or grayish green.

bottle-green adj.

{AHDEL}

In the thread Compounds and Phrases - differences, John Lawler (1) contrasts compounds with phrases, and (2) states that compounds are treated as if they were unitary:

Phrases belong to syntax. But phrases can be frozen, and those are dealt with as if they were single words.

Point (1) invites the question 'Where does the divide come?'; this has been the subject of various papers and is still disputed.

JL reinforces point (2) in the Labelling of noun components of a verb thread:

[S]cuba certainly refers to a noun, but it's been locked into the compound [in scuba diving] and isn't really functioning as a noun syntactically.

The sense in using this approach is endorsed by the fact that compounds with different variants (eg particle board / particle-board / particleboard; ink well / ink-well / inkwell) (but contrast blackbird / black bird) are really merely spelling variations.

JL does add that it is reasonable on the other hand to trace the etymology of compounds:

If you wanted to class these compound verbs [etc] as having nominal [etc] sources, that would be reasonable.

So 'interesting-looking', a compound premodifier and compound adjective, is an adjective. Classing 'interesting' as a (participial) adjective and 'looking' as a more 'verby' ing-form, we can see where the word originated.

The BBC World Service website has a good overview of the parts of speech that can be seen to have combined to form compound adjectives. But adjective plus adjective is also a possibility, as in 'blue-green'.