Learn English – Are focusing adverbs exceptions of adverb definitions

Searching Google Books, I find only 25 results for "helping-adverb" and eight for "helping-adjective," so I'm not sure if they were so much abandoned, as such, but, rather, never very common to begin with.

"Adverb of degree" receives 5,500 Google Books results (major contrast with the above), so I would certainly say that one would be better off, in terms of likelihood of being understood, if one used that term rather than the one with "helping."

In terms of "credibility," I don't see anything particularly wrong, per se, with the use of "helping-adverb"—just a lack of currency. "Helping adjective," seems wrong, however, as English adverbs are defined in any current dictionary or textbook as modifying either verbs or adjectives, and hence "helping adjective" seems to make little sense because all cited examples would normally be classified as adverbs or adverbial phrases.

The idea of a "helper" word is, however, used in grammar sometimes; the Chicago Manual of Style, for instance, mentions "helper verbs." However, since all the examples cited above do pertain to degree, one could argue that "adverbs of degree" is more readily intelligible.

I don't think any term directly translatable as "helping-adjective" is commonly used in describing Japanese grammar, but Japanese is an agglutinative language, meaning that multiple word-like entities are frequently combined together to form what could be considered single words but that do the work of what could be a whole clause in other languages. To wit, Ikaserarezu..., whose basis is the verb iku "to go," could mean, in context, something like "Without anyone forcing him to go..." The various things added to the end of verb stems and then onto one another in turn are usually called in English-language descriptions of Japanese grammar "auxiliary verbs" or "helper verbs."

The rub: some of these "verbs" are conjugated as though they were adjectives: -yasui, as in yomiyasui "easy to read" (from yomu, "to read"), is exactly conjugated like, and functions as, an adjective. In fact, yasui exists on its own as an adjective in its own right, etymologically related though with a slightly different meaning. Hence, one might (though typically doesn't) speak of "helping-adjectives" in Japanese, and these do not necessarily have anything to do with degree.

Korean and Turkish are other agglutinative languages, so somebody familiar with them might have more to add on your last question.

• 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.