It might be that you need to consume an amount of Vis every year to prolong the effect. I could see someone needing corpus Vis (which often is quite gruesome matter) to stay young
Not inconsistent, no. Just an incomplete description of the system. As it concerns the original researcher accomplishing the Major Breakthrough, AnM plainly states the discovery process automatically Initiates it, and that he needs to devise an Initiation Script if he wants to trade it to others. As it concerns the Cult Lore, I seem to remember that one of the authors of either AnM or TMRE stated on the issue that the research process counts as experience in the (Cult) Lore related to the discovery.
Yep, an Initiation Script represents an incomplete integration in Hermetic Magic. Stating it as partial and half-way is a correct description. A second Major Breakthrough would be needed for complete integration. However, if you look at it the other way, once non-Hermetic magic has been turned into a Mystery Virtue that is related to Hermetic Magic (either because it's an Hermetic Virtue, or a Supernatural one that can be used with Hermetic spells, e.g. Synthemata Magia), it has been partially integrated, so it's equally uncorrect to say it's (wholly) non-Hermetic, and has nothing to do with Hermetic theory. It's an hybrid, and surely there can be many possible Scripts for Virtues that have nothing to with HM at all (e.g. a Script to gain a general Virtue), but for ones that bestow Hermetic Virtues, or Supernatural ones that interact with Hermetic spells, rituals, familiars, enchanted items, anything else which uses Hermetic Arts, they do partake of the nature of Hermetic magic, albeit partially and peripherally.
That's another possible way to look at it, sure. The way I proposed allowed to acknowledge that magic which is at least partially Hermetic, such as Hermetic Mystery Virtues, can change a human into something else, without breaking the Limit of Essential Naure.
There might be groups interested in leaving the two major limits inviolate, even if after the revelation of the Mysteries (TMRE and HoH:MC) and Ancient Magic, it must be concluded that without massive houserulings of supplements, the vast majority of the minor Limits (e.g. Aging, Creation, Warping, Vis, Energy, True Feeling, Arcane Connection) are proven flaws in current Hermetic theory that could be remedied by integrating the appropriate insights from non-Hermetic schools. Bonisagus' work was good, but it is far from having reached its maximum potential.
Greater Immortality is another reality of the setting which forces to conclude that either Essential Nature, too, is a flaw, or there is more hidden features to human nature that can be safely concluded from studdies of mundanes unexposed to magic and supernatural powers.
Your reasoning has merit, but it needs some more examination of the meaning and scope of Breakthroughs.
By examining which ways the Mystery Virtues in TMRE and HoH:MC, and the possible breakthroughs in AnM, challenge the standard Limits, I think a reasonable guideline which is true to the books might be: a discovery which breaks a Limit but in a rather specific and specialized area, pushes the limit but does not break it, hence is a Major Breakthrough. A Discovery which does away with all or most of the Limit would do an Hermetic Breakthrough.
E.g. The Parma Magica completely abolishes the Limit of Universal Magic Resistance in all ways and applications that matter and so is an Hermetic Breakthrough. Integrating Hyperborean Magic to create vis-less creations or Defixiones to create unlimited spells does away with most of the Limit of Creation or Arcane Connections, and so they are Hermetic Breakthroughs, too. Conversely, Hermetic Mysteries like Hermetic Alchemy, the Heart-Beast, Glamour, do break some Limits but in a rather more specific or indirect way, so they fit for the definition of Major Breakthroughs.
Following this, if we examine what the various Greater Immortality methods do, we can see that each of them does allow and change the nature of a human in a rather limited way, only a specific kind of immortal supernatural creature per Virtue. Hence it fits the definition of "pushing" a Limit. The Hermetic Breakthrough equivalent for this field of magic would the invention of a new laboratory activity or ritual spell type that would allow mages to turn humans into any kind of non-divine supernatural creature, humanoid, beast, spirit, ghost, and do the reverse, too, creating true humans from beasts and spirits, returning ghosts to true life, etc.
Hmm, I honestly think that the way you apply the definition of Hermetic Breakthrough here is far too strict. A HB means a previous Limit is really no more a worry at all for mages in all or most of the magic uses that are related to it. Ie. Limited Magical Resistance ? A worry of the past, thanks to the Parma. Arcane Connections ? A worry of unenlightened times, once I discovered Unlimited Range spells. So-called breaking the Limit in one specific application among many is no breaking at all, only pushing the limit. E.g. discovering a quasi-Parma, that only works against all Infernal powers.
Are potentially.
IIRC, the mysteries, and even more ancient magics, are optionnal.
I seem to recall a phrase specifically telling something like "not all ancient magics are good for all campain" or "should exist in the same campain".
Thus, the fact that there are suplements describing ways to do away with most of them tells nothing about hermetic magic, since these might not exist in a given campain
Hum... Another possibility might be that the various method "kill" you, transfering your spirit into a construct of sorts, even if it appears like you've just changed.
Yes, of course. However, the fact such potential does exist makes all the difference when one evaluates the nature of Hermetic magic.
That is nothing more than standard caveats from the author covering the back of those SG that do not want to use some of those rules in their campaign. However, when making an objective RAW evaluation about the nature of Hermetic magic as a whole, the only workable way to make such is to take all published rules at face value, unless some are explicitly told to be incompatible, which the bit you quote is not. Besides, mysteries and hermetic research are so entrenched in the setting as well, that expecting them not to be used or at least potentially valid cannot be the default. Maybe not all mystery cults are present, maybe not all ancient secrets are discovered, but the fact is, Hermetic magic is not expected to have reached its maximum pontential for development by the standard present in the corebook.
Heh, by the same standard pretty much none of the Limits of Magic might ever exist in a given campaign, the Limit of the Divine is the only one whose absence would put the metaphysics of the setting in serious question.
All House Rules (and Rulebooks) Are Created Equal. 
Conceded, this is a valid theoretical possibility. However, it is one that I would kindly urge you not to touch with a 10-feet lead pole, as it opens up a terrible can of worms about the identity of the characters.
Since the game so far wisely shuns from ever having the mind/spirit/soul unity of the characters ever broken (except maybe in the wholly undefined and fuzzy afterlife, which the characters are really powerless to affect or investigate), it is safe to assume that even in the ArM universe, the real "you" always goes where your consciousness and self-identity goes, and if you think you are Tyranus of Bonisagus, and you have his memories and personality, you are him for everything that matters. The soul is thankfully kept offstage in the game, and until the character meets final death, there is no reason to assume it does not always go where spirit and mind go.
I would also point out that for characters to whom shapeshifting may be a rather trivial matter, using the body as the main benchmark of identity is rather unwise. Also, there is nothing in the metaphysics of the setting that prevents the soul from staying with the rest of the person in a immortal form, until Judgement Day.
That's somewhat problematic as these points are already used to form the Supernatural Ability, if any.
I think at least some of the Ancient Magic examples, however, cultiavted a Lore score in addition to making the magical dicosveries, so this is certainly a viable path. Another is to require the development of such an Ability from scratch, based initially on simple mundane qualities of the cult and its non-working rituals, and only later extending to mystical teachings once the first script is invented; I believe TMRE mentions something like that.
Yeah, you make a good argument that this seems to be the line's policy on the interpretation of Major/Hermetic.
I agree completely that the rules for developing Breakthroughs are far too easy. The number crunching for acquiring Stabilisation points toward a Breakthrough can be borne out. If House Bonisagus actually cooperated on a few of the Hermetic Limits, they should have surpassed them long ago.
A "house rules" answer is to treat the objective numbers (30, 45, and 60) as the BASE target. Add a Simple die to Minor Breakthroughs, a Stress die to Major, and the product of a Stress AND a Simple die for Hermetic Breakthroughs. It'd slow down the discoveries, but not prevent them.
Arguably, the Limit of Warping is the one which Magi should go after, first. It is the true "ultimate limit". Once it has fallen, there is a much lessened risk for engaging in Stabilization of spells in original research, and development would accelerate.