Are there any existing rule systems that allow faeries (true faeries, not Merinita weirdness) to use, and teach, Magic-realm abilities (often abilities requiring the Magic-realm Gift)?
I ask because it seems like there's a major hole in the setting when the two are separated. Or to put it more crudely: any rules that indicate that Odin cannot use (and teach) Norse Rune Magic are bad rules. 
Yes. Take a look at Homunculus Wizard in RoP:F.
Also see faerie teacher, though that isn't quite what you are looking for with Odin...
also a faerie can act as a mystagogue, probably sending people on quests in order to obtain wisdom... it seems like a natural role for a faerie.
Indeed. I may also need to permit a Faerie version of the Gifted Quality, for situations where a Faerie's role requires it to function as a sorcerer in a specifically Magical tradition - Odin and possibly Mercury are the only clear-cut examples that spring to mind, however. The rule that "Faeries may not have supernatural powers associated with the...Magic Realm" can get bent.
Gruagach can be represented with a Grant Virtue ability, I think - using Grant Virtue for an Opening makes a certain amount of sense.
Insisting that Odin be a faerie akin to those of late Medieval fairy stories is the bad rule here.
I personally like the idea of faerie versions of Gods being based on more ancient magical versions, but that doesn't really work well with the Gruagach tradition... which really I feel someone should have questioned that design before it was published...
Disclaimer: I really like Faerie mechanics in RoP:F.
That said, I don't think that rules that allow Odin to create (faerie) supernatural effects that appear to be Rune Magic would be bad rules. Surely, Odin knows Rune Magic better than any mortal magician to whom he can teach it...
As for teaching Rune Magic, this can be read as one of two rather different tasks.
The first is teaching runecraft to someone who does not know it -- in game terms, opening someone's Gift to viktar magic and giving a score of 0 in every rune. In this case, I doubt Odin would do it in the same (unepic) way as any common Viktir master would. Odin would say "Well, here's how I did it; I hang for 9 days etc. etc. You can do the same!" The rules for that (including help that can be obtained to survive the ordeal) can be found on p.124 of HMRE.
The second is teaching more runecraft to a Viktir who already knows the runes. This can take the form of Granting Puissant (Rune); Granting some other Rune-related Virtue; Granting a Faerie Power with use of a "new" Rune as a Charm; acting as a Faerie Instructor (RoP:F, p.62); or helping the character retrieve achieve the same benefits "on his own", or gain some other Study Source, by guiding him on a quest in Elysium or Eudokia.
But could a Faerie taking the role of a hermetic magus teach Parma Magica?
^^this^^
Remember that there is Odin the Magical being who does stuff and the Odin the faerie representative that generates stores about Odin so it can gain substance. There is no reason why the former could not teach Rune magic and the later just take credit for it.
I like that variant.
Except that the rules in Rivals indicates specifically that Norse deities are all faeries who are opposed by the Magical Jotun...
which means that according to RAW, there is no magical Odin. Which in my opinion was a mistake.
So ignore the fluff. What I proposed is MY way of viewing the Gods vs faerie issue. Quite simply, there are or were magical beings who performed the actions of the oceans or lightning/storms but the faeries have stepped in front of them to suck up the attention that the magical beings do not really care about. Both exist. It is much more likely that a human has interacted with a faerie representation than the actual magical one.
Or perhaps the magical being remains the significant one and the faerie is the protagonist of exceptional stories. I can definitely see potential from the "faeries pretending to be the Old Gods" idea but it doesn't take much repetition for this to wear thin for me.
If the Magical power of the Old Gods wears thin, e. g. because their sacred sites are destroyed, the faeries impersonating them take over. That's the way I like fully immanent religions represented in our Mythic Europe campaigns.
Some Old Gods in ArM5 have been exclusively faerie all the time, following the preferences of some Tribunal book's authors. LotN has Egyptian cults both ways. But RM's primal age Odin was the child of Urdur, hence a Magical being (RM p.75), which also in ArM5 canon allows to represent his sacred sites as Magical.
Cheers
And on the same page it describes the Jotun as being primal magical being bound by the faerie Gods who are the aesir, which would include Odin.
I appreciate the inter-generational rivalry of the gods being included in Ars Magica, but the idea of it being Magic versus Faerie was a poor choice, IMO.
As David Chart explained in subrosa #16 p.5, the ArM5 authors made the Aesir faerie gods in 1220 Mythic Europe, because at that time people did no longer believe in them. RM p.75 box represents The Mythology of the Muspelli, not canonical in-game truth.
You can see from the text of RM p.75 outside that box, that in the primal age the ArM5 Odin of Germanic pagan culture was a Magical being. This can well have remained so, while that pagan culture existed. But this did neither prevent faeries from pretending to be him, nor jotnar - "not necessarily the most neutral of witnesses" - from exploiting such impersonations.
Cheers
Actually no, I don't see that. I see that you have inferred that from the text, but I do not see that it is necessarily so.
Now you write:
To sum it up:
By RM p.75, the primal age ArM5 Odin of Germanic pagan culture as the son of Urdur Fate-Spinner, the jotun of fate (RM p.81), necessarily was a Magical being. So RM does not "indicate specifically, that Norse deities are all faeries who are opposed by the Magical Jotun". To arrive at such a claim, you had to read RM p.75 box The Mythology of the Muspelli out of its stated context as an account by "not necessarily the most neutral witnesses".
Cheers
If I were to assume you were the authority on such context I might be persuaded by this insult. I do not. I have to leave for a 6 hour trip to pick up my children soon so I am not going to dig through the text to argue with you at his point about 'proper context'. To my reading it seems clear that the Aesir are declared to all be faeries which enslave the Jotun and make use of their powers as described in ROP:M. It is at minimum nowhere near as clear as your assertions would indicate that Odin was magical. Depending on which version of ancient history in Ars Magica you wish to utilize, Odin may have been a magical being which became Faerie as he embraced worship to enhance his power, may have been a faerie changeling child of a magical being, or may have been a magical being which was replaced by a faerie when his legends grew large enough. Frankly none of these fit well with Norse mythology and the character of Odin, but these are the messes that get left behind when RAW is a hodge podge of unconstrained fan writings with a lack of direction behind either a unifying vision or a commitment to keeping the game general enough to allow people to play it their own way, instead relying on house rules.
The context of RM p.75 box The Mythology of the Muspelli is obvious to those able to read that box. There is no authority needed to explain that this box does contain - yes - The Mythology of the Muspelli, who are "not necessarily the most neutral of witnesses".
Indeed, RM leaves such options open to the troupe - but for the one with Odin a faerie changeling.
Cheers