I have read conflicting views on whether Ein Sof (אין סוף) is a name of G-d, like for instance Havaya, or it describes, so to say, the Atzmuth. For the Gra and the Ram'hal, it seems that Ein Sof is a name. Ramhal is explicit about this in Adir Bamarom (and maybe also in Kala'h Pit'hei 'Hokhmah). The Gra suggests this view as well in the Likutim at the end of his commentary on Sifra DeTsni'uta. However, for the Ramak, and for the Arizal, I'm not sure it is true. I would like to have a clear picture of the different opinions, and the reasons for the divergence of opinions.
The most authoritative written source on this subject is Sefer Shorashei HaShemot by Rabbi Moshe Zacuto z”l of Mantua in Italy, which lists all the Holy names and their particular qualities as found in the entire written Torah and Nach.
In section Aleph, the term Ain Sof does not appear. That means it is not a Holy name, but a description of a concept.
Regarding the portion of your question directed to the subject of G-d’s essence (and being), both Kabbalistic teaching and Chassidic teaching explain that G-d at His essence transcends all names and letters. There are numerous places that discuss this like for example Sefer Emek HaMelech and many others.
Concerning your claim that Ramchal in Adir B’Marom says Ain Sof is a name, he states there precisely the opposite. See page 91 of the 1991 Jerusalem edition ספר אדיר במרום השלם which says
ועל כן אמר אליהו ז״ל בתפילתו, שלאין סוף ית״ש לא נמצא שום שם, כי אי אפשר לגדור אותו בשום שם.
The usage of blessed be His name does not mean Ain Sof is a name, but rather that G-d possesses names in order that His creation can relate to Him in some capacity. The name is only for the other and is not His essence.
My personal tradition from my Chassidic (and Kabbalah) teachers, is that "Ein Sof" is not really a name of G-d or a description of "Atzmuth" ("The Essence") at all.
The Hebrew words "Ein Sof" simply mean "without end".
Kabbalists did not wish to describe G-d by what we think He is, (in defining terms) because any description of G-d by humans would fall short of being exact.
Even the term "Atzmuth" (Essence) is simply stated and not described.
So the next best thing is to describe what He is not. The only thing we can surely say about G-d is that He has no end (no limit).
I hope this helps. :)
Hashem has only one proper name: Y-HV-H.
Everything else wanders into description. For example, Elohim also refers to human judges.
Still, some of those descriptions are considered specific to qualify as names when it comes to the prohibition against erasing the name of G-d. The Rambam has a list of these names, (Mishneh Torah, Yesodei haTorah 6:2), and the Shulchan Arukh quotes the Tur's slightly different list (Yoreh Dei'ah 276:9).
"Ein Sof" isn't on those lists.
It isn't even a description of the Atzmus (Essence), as no such description is possible or apparent attributes of G-d. It is a description of what He isn't. Thus "Ein Sof -- Without End" -- "In+Finite".
In that sense, I think the Rambam would have liked "Ein Sof" as an appellation for G-d, had he been exposed to it. The notion of calling G-d by what He isn't fits the Rambam's notion that all of the so-called Attributes of G-d are either (1) descriptions of how His Action appears to us ("Merciful" -- he acts in a way that a human would when feeling mercy), or (2) descriptions of what He isn't ("Eternal" -- isn't subject to time). The Rambam literally compares the notion of attributes of G-d Himself to Christian trinitarianism (Guide to the Perplexed 1:50):
Those who believe that God is One, and that He has many attributes, declare the unity with their lips, and assume plurality in their thoughts. This is like the doctrine of the Christians, who say that He is one and He is three, and that the three are one. Of the same character is the doctrine of those who say that God is One, but that He has many attributes; and that He with His attributes is One, although they deny corporeality and affirm His most absolute freedom from matter; as if our object were to seek forms of expression, not subjects of belief.
But that's not distinctly Rambam, who isn't really a primary source on Qabbalh issues. The same point is made by the Vilna Gaon (Asarah Kelalim, kelal #1, opening paragraphs). None of Qabbalah describes the Ein Sof itself, as that is inherently indescribable. Which is why we refer to it in the negative, with the word "Ein". We are saying what Hashem's Essence isn't, not what it is.