So the Limit of Essential Nature doesn't prohibit using magic to maim an enemy and at the same time doesn't rule out magically healing that effect. That's something to keep in mind if you allow a spell to destroy the Gift.
Magi seem to have difficulty affecting the Gift, as exhibited by their inability to reverse its social effects. It may be that Hermetic Magic is unable to destroy the Gift not because of any magical Limit but simply because of flaws in magical theory. It seems like a troupe could easily go either way on this one.
It's a good question whether the process of Mystery initiation actually alters essential nature. Breaking one of the two fundamental Limits is a big deal in the AM5 universe. Certainly there are a few individual mysteries (I'm thinking of the immortality ones) which break this Limit but that's different from the basic process of entering the Mysteries having such a far-reaching effect.
It absolutely is, because the spell in question (Embrace of Boethius) is not a Mystery effect - it can be cast by any magus who knows it, or else it would say "this spell requires the Verditius Outer Mystery to cast." Ergo, Hermetic magic can intentionally damage the Gift.
The spell is required for the standard Verditius initiation into the House's Outer Mystery, but that's its only link to the Mystery. It isn't even required to be used if you want to initiate Verditius Magic (if you don't mind being Wizard's Warred into oblivion over initiating the Mystery without the Gift flaw).
I'm pretty sure Ordeal Flaws do alter Essential Nature. So there's that.
"Initiating a Mystery alters a person's Essential Nature" is consistent. There's no Hermetic magic that will initiate a Mystery for you - indeed, Mystery initiation can be explained as "the process of refining and altering one's Essential Nature through interaction with one of the Four Realms, most commonly Magic." So Mystery Cults are not 100% Hermetic.
Of course, it's also possible that Mysteries are simply a form of Warping the self (which is a Lesser Limit rather than a Greater and may be more flexible).
Greater Limits supposedly apply to all magic, not just Hermetic. I can think of a few exceptions - the Immortality Mysteries and the permanent transformation of Mythic Alchemy and Solomonic Alchemy - but these are all difficult and require Vis. Anyone can do an initiation at no particular cost. It seems to easy a way to break the Limit.
I disagree, but this is subjective. I personally think that the concept behind Mysteries has similar symbolism to alchemy, and amounts to "growth through transformation," and more to the point I don't have a better way of explaining why the Flaws are unremovable - and it does fit for Mysteries to be a kind of true transformation.
(Also, point of order: Solomonic Alchemy doesn't require vis for its permanent effects. It does this through imitation of natural processes, which is a way of saying that it's a really weird form of Rego-craft that doesn't violate any Limits. It tap-dances on the line of Creation, but Hermetic Rego crafting can do most anything Solomonic Alchemy can do, if you have enough Finesse.)
Solomonic Alchemy mimics both Philosophical Alchemy (from A&A) and Mythic Alchemy (from Hedge Magic). The enhancement guidelines, analogous to the A&A skill, do not take Vis. The transforms mimicing Mythic Alchemy are ritual guidelines and so do. I think of it as akin to the difference between Rego and Muto.
Bummer for Solomonics then, they're even worse than Mythic Alchemists. Although Cradle & Crescent is one of my favorite line books overall I never gave too much attention to the Sahirs, as I prefer the 4th edition version of them.
In this case, the list of violations of the Limit is even shorter - Mythic Alchemy, which is cumbersome, limited, and extremely vis intensive, and the extremely rare immortality Mysteries, which also cost vis. At least the ones in Mysteries do.
A note: Embrace of Boethius does not affect the gift, but the mind, destroying certain aspects of hermetic understanding. A spell which removes all knowledge of magic would not quite be the same as destroying the gift, but would be equally intimidating, especially if it results in massive warping the same way that Embrace does...