Idioms are NOT idioms??

Much better, now I get what you are asking about and I will try to answer your question .



As for the first thing:

This is true. Specifying, in this sense idioms are proverbs, catchphrases, figuratives, etc. Example of English idioms:
Proverb: "Birds of a feather flock together."; "Absence makes hearts grow fonder."; - Here, the meaning is implied.

Catchphrase: "An apple a day keeps the doctor away"; "This is just one example off the top of my head." - You don't literally mean that you have this example somewhere near the top of your head. What you mean is that this is the first example that you have come across.

Figurative: "Everything was ok, but the penny has dropped when he said his last remark." - Here, what you mean is that He said too much and too inadequate. You can also say the same with: "He has gone over the line". None of these two you mean literally.



As for the second thing:

These are not idioms as such, but this is the IDIOMATIC language. The aspect, which you have given in the quote above, is called in grammar "the rection". Rection of a noun determines with which prepositions the noun works, and with which it does not work. Using your example: "the ability" will work with the preposition "to" + do sh. But it will not work with "for/of" + doing sh. Why? Just because. The language has evolved to this form. It could have been else 100 years ago, and it could be else in 100 years. There is not any rule that would describe or group the rection in some general way. You simply have to memorize it. Nouns, verbs, and adjectives - every single one of these parts of language has its own rection.

Why these two somehow different aspects of language are described as idiomaticity or idiomaticalness?
Because these are just fancy words for a very simple thing: when something is idiomatic, it essentially means that it is native to the language. Idiomatic language sounds natural.

For example, in English you say: "This is just one example off the top of my head."
Whereas if you wanted to say exactly the same thing in Polish, you would literally say: "This is just the first example from the edge" (pol.: To tylko przykład pierwszy z brzegu). It does not sound naturally in English, but it does in Polish. So, this is a Polish idiom, or, better, the polish version of an inter-linguistic idiom.

In English you say: "He is all alone."
And in German the same thing you express by saying: "He is lone as mother's soul." (de.: Er is mutterseelenallein). In English it does not sound naturally. In German, it does - it is the German version of this idiom.

I don't know if there are any idiomatic phrases exclusive to only one language. This is rather irrelevant.

Referring to the GMAT, I think that the GMAT does not test English proverbs or figuratives. The GMAT does test some English catchphrases, as they are much more common in everyday language than proverbs or figuratives. The GMAT does for sure test the rection: as a matter of fact, this is what the sentence correction is mostly about. Rection is very important thing on the SC part, and you can lose or gain many points on them. One list for idioms is contained in Manhattan SC, Chapter 9: Idioms.

Does it ring the bell now?