What say you, does a current apprentice count as an AC to their Master? Or vice versa?
No.
Nope. Appropriately workaholic ones might be ACs to the lab, though.... 
Yes and no. There is a target, "Family line", yes? So the "Ties that bind" are arcane connections. Making them Arcane Connections will require a small breakthrough.....
Where is that Target? The Faerie Magic Target "Bloodline" doesn't work that way. It only hits descendants, not ancestors.
I'm not sure that works; Arcane Connections are naturally occurring phenomena, so I don't see how a Breakthrough in Hermetic Magic would change which things are ACs. Unless you mean making a Breakthrough in the Opening of the Arts that ties the master and apprentice together. But who would actually want that?
Hello, Callen. My thinking seems simple to me. The target "bloodline" establishes there are arcane connections based on things other then physical connection. In that case, the connection is "related". It cannot, as written, go "up" the bloodline, but arcane connections go both ways, by definition, and some research should make it possible. Similarly, there is a bond between a Master and his Apprentice. It might be love, it might be hate. But there's nothing in the rules that says a emotional bond can't be an arcane connection, and quite a bit of precedence for it ("Bloodline", Familiars, True Love, etc). Should be possible, with reasearch, to access that, turn it into an Arcane connection.
Bjornaer magi, at least, create that kind of connection, as per page 18 of HoH:MC, in the upper right corner.
If any fixed Arcane Connections to Birna still exist, they must represent an enormous danger to the House in the wrong hands, assuming she still exists as a Great Beast. A Range : Arcane Connection, Target : Bloodline spell with such an AC could in theory target the entire House. OTOH, House wide spells/rituals that are tailored to the Bloodline for Warping purposes could be both vis-efficient and just plain scary.
Perhaps, but I can't follow what seem to me to be several leaps of logic that you use:
How does Bloodline establish this at all? Faerie magics can affect things via sympathetic connections, too. You have to establish why this is an Arcane Connection rather than some sort of Sympathetic one.
No, they don't. They may generally go both ways, but certainly not by definition. There has been disagreement on this in the past. When I read the ArM5 p.84 description, it certainly sounds like the connection had to be part of the target in some way, and generally two things are not both parts of each other. So I tend to read that as they are one-directional unless otherwise noted.
Much of that "precedence" is not precedence at all. Bloodline doesn't require any emotional bond at all. They need not know anything at all about each other, and it will still work. A familiar whose bond is severed (if it survives) will forever have an emotional bond but won't be an Arcane Connection forever, decades at most. I haven't found the statement for True Love, but even then True Love is something greater than just any emotional bond, after all Hermetic magic can do a lot with emotions while True Love is beyond it.
I think you skipped right over the logic. Is there any reason Sympathetic Connections cannot be Arcane Connections? Looking over the Limits of Magic, I would say no. Again, I am not saying it is an Arcane connection, I am saying it could be, with Research.
No. To put it bluntly, this is not correct.
It it not only not true by definition, but not even generally true.
So a toenail is an Arcane connection to you, but you don't have an Arcane connection to your toenail? Huh.
They already are, by definition, but Arcane Connections are a specific form of Sympathetic connection that is stronger than any other Sympathetic connection.
But quite simply, you can't use other sympathetic connections unless you have an arcane connection, and being a relation, bloodline, or part of a hermetic lineage is not a valid connection. Also, I don't see any mention of where True Love is an AC to the other person. Familiars are ACs to their magi, and vice versa, because of the binding ritual.
Lucky Mage makes a valid point, that you're trying to use Original research into a tool that allows you to turn something that isn't an AC into an AC. You could probably do that, but I don't think it's RAW, and mentioning Original Research as an avenue makes it well outside the bounds of RAW, since Original Research is about changing the rules of magic beyond what the rules presently allow. In other words, troupes can do whatever they want with Original Research. Offering it up as a possible reason in defending your position is a weak defense.
And Tellus has it right about the directionality of ACs, only two that I can recall off the top of my head: the talisman, and the familiar are bidirectional ACs. I think callen might be aware of a third, but I always seem to forget it.
Yes, absolutely.
I could have swore there's a spell in canon that destroys Arcane connections to you. Sigh. And there's that spell that moves your image as well.
Oh there are spells that destroy ACs. They don't do it at R:Arc.
To do so automatically, at R: Arcane, has been a grail sought by many, found by none.
No, really, I didn't skip over it at all. I did not argue Sympathetic Connections cannot be Arcane Connections, and I won't since they can be. I did not argue there is no issue with the Limits of Magic. I did not argue you could not do Original Research to work on it.
If the entirety of your argument had been "it's not against any of the rules, so you could potentially do Original Research," I would not have objected at all.
But you presented your reasoning, and within it presented several things that were either false or do necessarily follow by logic. I objected only to those.
In my sagas, I wouldn't even accept that as Original Research; Hermetic magi don't get to decide which things are Arcane Connections any more than they get to decide which tethers are resonant enough to make auras. But, well... I guess that's my saga, not Saxonous's. shrug
Which is why I suggested Original Research isn't really a strong argument. OR is the means by which the troupe agrees to extend the rules of Hermetic magic for a given saga. There are several suggested breakthroughs but that is all they are, suggestions.