When did Kurds and Persians first names use ?

I am sorry but Assyrian language was already extinct by that time. Adiabene is founded in 15, whilst the spoken Assyrian Akadian most likely began to completely fade away before the 4th century BC. Its total extinction is based on the last text pertaining to it which belongs to the fist century. The official (not native) language of Adiabene was Aramaic, which brings no wonder nor any point about since this Semitic language was already the lingua franca and an official language of the whole Iranian empire. Even there are 1000 Aramaic words (known as Hezarvash) which were used in the official letters written all over the territory of Iranian languages (Sogdian, Parthian, Middle Persian, etc.). Also it had a huge impact on Semitic speaking folks of Mesopotamia to the degree that even Jesus Christ as a Jewish by blood spoke Aramaic for sure (with the appearance of Adiabene simulatneously).

I doubt it at all. For your knowledge the first and the only time that the term Assyria appears in the history of Adiabene was when Romans conquerred it and named it Assyria. Afterwards some other western historians followed such a Roman designation, which does not match with the facts that we know about Adiabene for sure:

-Its ethnic majority is uncertain up to now but we know that Adiabene was socially a mixture of various people.

-There was most likely an Iranian ruling class over there since its rulers' forenames are to be explained by Iranian etymology (e.g. Izates, Meharaspes, etc.).
 
-Its official language was Aramaic, like the other western parts of Iranian territories (since the reign of the Cyrus the Great who put an end to the Babylonians and ancient Assyrians bestiality and respective languages forever and made Aramaic as the official langauge and lingua franca in the entire mesopotamia, though it was later replaced by Arabic in the post-Islamic era).

-Romans conquer Adiabene some 100 years after its appearance in the history and then name it Assyria after its annexation to their province of Assyria.

-Afterall Arabs invade Adiabene and there they only face "Al-Akrad", which is Arabic for "The Kurds" indeed.

-After the collapse of Adiabene, no other people ever declared to be related to it, but Kurds did in fact. Hedhbani Kurds, whom Saladin was their descendant, are named after Adiabene.

-Adiabene fame could be speculated mostly due to its conversion into Judaism and the later assistance that it contributed to the other Jews in their fight against Romans. As a matter of fact today all Jews from the respective area do consider themselves ethnic Kurds and not Assyrian, however their religious language is-the same as the bulk of Middle Eastern Jews, an Aramaic speech. Their respective Neo-Aramaic speech is even referred to as "Kurdit" by the other Jews in fact.

It firstly converted into Judaism very soon after getting established which paved way to the later conversion into Christianity subsequently. By the way you cannot overlook or deny the existence of Judaism and Zoroastrianism at the same time over there.

Any accounts? By the way what are the elements of your ancient Assyrian culture that any Iranian folk could ever adopt?

As I have already said the first and the last time that the term Assyria was applied to Adiabene was after its conquest by Roman invadors in 115-116 when they annexed it to the Roman Province of Assyria. I am sure you are aware of the fact that the Kingdom of Adiabene became over in the very same year-116!

I am not gonna doubt your know-how, but in terms of Iranian linguistics you seem to have a vague idea. The Scythian essence of Kirmanji Kurdish may even be much clearer than that of Median in fact. Since we only know a few words from Medes' language which are mentioned within the ancient Greek accounts whilst on the other hand there are a pretty good amount of ancient and middle Scythian (particularly Khotanese Scythian) materials at reach (e.g. retention of ancient "c-" in the common Iranian verb for "to go" is exclusively restricted to Scythian, Kurdish, and Ossetic: Old Iranian "cyev", Avestan "shyev", Parthian "shew", Gilaki "shu-", Azari "sho-", Middle Persian "shew", New Persian "sho-",  Scythian "cu-", Kurdish "cu-", Ossetic "cu-".) 

For your knowledge, again, the case of being western, eastern, southern, or northern, is merely theoretical in terms of Iranian linguistics. For instance Kurdish as a theoretically Northwestern Iranian language has been disputed to represent Southwestern characteristics (despite whether the conclusions are reasonable or not). Linguistic theories are to be alterred or refuted based on rational reasonings out of authentic materials.

Indeed the essential likenesses between Kirmanji Kurdish and Ossetic outnumber those between Kirmanji Kurdish and Persian (I am sure you are not gonna reckon loans into similarities anyway). I myself am conversant with Persian and Kurdish languages as well as am a linguistic buff, particularly regarding to Iranian linguistics (which of course encompasses Scythian and Ossetic).

Not at all.

That is definitely wrong. Historically and culturally, Kurds do share their very exclusive part amongst Iranian peoples.

In accordance with the following genetic researches Kurds are:

"Similar to Azeris [do you need to be reminded of Azeris Iranian origins?], Ossetians [an Iranic people indeed], and Armenians" ~ Richards M, Macaulay V, Hickey E, et al. (November 2000).

"Possibly closer to European rather than Caucasian." ~ Nasidze I, Quinque D, Ozturk M, Bendukidze N, Stoneking M (July 2005). "MtDNA and Y-chromosome variation in Kurdish groups". ~ Annals of Human Genetics 69 (4): 401–12

"Sharing close similarity with Georgian in terms of MtDNA sequence." ~ Comas D, Calafell F, Bendukidze N, Fa?an?s L, Bertranpetit J (May 2000). "Georgian and kurd mtDNA sequence analysis shows a lack of correlation between languages and female genetic lineages". American Journal of Physical Anthropology 112 (1): 5–16.

"Originating from the same Old Mediterrenean ancestry along with Turks, Iranians, Jews, Lebandese and other Western and Eastern Mediterrenean peoples." ~ Arnaiz-Villena A, Karin M, Bendikuze N, et al. (April 2001). "HLA alleles and haplotypes in the Turkish population: relatedness to Kurds, Armenians and other Mediterraneans". Tissue Antigens 57 (4): 308–17.

"Close to  Anatolian Turks along with the other East Mediterrenean peoples" ~ Arnaiz-Villena A, Martinez-Laso J, Alonso-Garci? J (September 2001). "The correlation between languages and genes: the Usko-Mediterranean peoples". Human Immunology 62 (9): 1051–61

"Relatedness with Jews [by blood]" ~ Nebel A, Filon D, Brinkmann B, Majumder PP, Faerman M, Oppenheim A (November 2001). American Journal of Human Genetics 69 (5): 1095–112.

Accordingly there is definitely no such a point in regards with genetics to dismiss the Iranian essence of Kurdish people.

Also I didn't notice a single direct mention to Assyrians within the above sources, as well as you had omitted Iranians obviously. Although I believe it is not intentionally.

That is wrong. If you once excluded Southwestern loans, there would remain no exclusive similarities between them both unless some dialectal likenesses which are hardly to imply any significant connection out of the theoretically common Northwestern Iranian characteristics (e.g. adding an initial "h-": Kurdish "hesin" ~ Baluchi "hesin", Kurdish "hesp", Baluchi "hesp"; which also is noticeable amongst other Iranian languages indeed and thus not that much remarkable).   

And it has nothing to do with the fact that Kurds are an Iranian people, unless fallaciously. I recommend you to observe various aspects of Kurdish people (their culture, their feasts, their calender, their months, their lifestyle, their proverbs, their myths, their folklore songs, their costumes, etc.) and compare them with those of the other Iranian poeples in order to appreciate the very Iranian tone of Kurdish people in practice.