Learn English – what word class do the words ‘so’ and ‘really’ belong to? (intensifiers or adverbs)

One problem is that the entire concept of "part of speech" is very old. How we use it in English, especially in dictionaries, goes back to the study of Latin and Greek. In this view of English grammar "adverb" is the catch-all category where everything that doesn't fit into one of the other traditional categories ends up. (The others being noun, verb, adjective, pronoun, preposition, conjunction, and interjection.)

Now there is no one, true description of any language (except perhaps constructed languages such as yours). There are merely alternative or competing descriptions which appear over time as more independent analyses of the language are undertaken. Such descriptions or analyses may be called "grammars".

Most (but not all) grammars include a concept of word class under one name or another. So one problem is that "part of speech" has two meanings. One is the specific set of eight categories from the classical languages, the other is as a synonym for word class, which is a lot looser.

So all your example words are adverbs under this older stricter view of parts-of-speech, but their qualities and quirks can be much more thoroughly investigated in newer ways. And various new ways will have various new terms for the classes they put these various words into.

Unless you are inventing a new language specifically to embrace the classical parts of speech you don't have to worry in which they belong, but if you are inventing a new language to learn more about how language works then it will be worth your time reading up on the many newer grammars and language descriptions and analyses.

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.