Are the Yomim Noraim UNIVERSAL days of awe or are they parochial and specifically Jewish?
They are both [typical Jewish answer]. On Rosh Hashana the ENTIRE world is judged by Hashem but Yom Kippur is a special day JUST for the Jews. There are two covenants Hashem made with the Jewish people according to Kabbala. One is the ברית המעור - the bris mila, and the other is the ברית הלשון - the convenant with our speech. These two covenants are represented by Rosh Hashana and Yom Kippur. Rosh Hashana is the ברית הלשון which is why we blow Shofar as sound is the root of speech and at Sinai we heard a Kol Shofar at which time we were given the Torah which is learned by mouth. Yom Kippur is the ברית המעור - that special seal which was given only to the Jews. In fact, according to the Medrash, Avraham Avinu had his bris on Yom Kippur.
The pasuk says כי קרוב אליך הדבר מאד בפיך ובלבבך לעשותו - The matter is very close to you, in your mouth and heart to perform. The "mouth" is the ברית הלשון while the "heart" alludes to the ברית המעור [as the pasuk says ומלתם את ערלת לבבכם - the heart requires circumcising].
This reminds us of the Torah that was first given as the aseres hadibros with SPEECH and then as luchos which were INSCRIBED with the words of Hashem [as the bris is inscribed upon us]. A hint to this is in the pasuk concerning Yom Kippur מכל חטאתיכם לפני י-ה-י-ה - The last letters of those four words spell מילה.
That is why the second luchos were given on Yom Kippur which is a seal on the hearts of the Jewish people and is like Olam Haba which is why we do it on the eighth day which represents Olam Haba. Yom Kippur called שבת שבתון is also a taste of the world to come. No eating or drinking - just basking in G-d's light.The Satan takes the day off [השטן = 364 but on the 365th day he is off duty. That day is Yom Kippur] just as in the future there will be no evil.
The pasuk says אנכי אנכי מוחה פשעיך - I I erase your sins. Why the double אנכי? One for the aseres hadibros which were verbalized and the other for the luchos [representing the ברית הלשון וברית המעור as we explained]. The more we immerse ourselves in the Torah which is called water, the more our sins will be erased.
[Based on Sfas Emes שנת תרנ"ו - There is SO MUCH MORE. עיין שם.]