Cracking a few more pots and throwing random ideas around, could Symeon's Daredevil fighting abilities been due to seeing the world through warging? A skill he presumably kept secret for fear of prejudice and to keep his tactical advantage (and because it's just that much more badass if people think he's simply that intuitive), all of which would link with the sapphire=secret stuff.
Arya then later uses this exact same method to get intel while she's temporarily blind.
Not saying it's any more plausible than the sapphire eyes being Other eyes or part-Other eyes, it's just another idea that came into my head when reading this very interesting thread and which I thought should be thrown out there.
I think it's an interesting question whether Petyr's kiss was deliberately done to provoke Lysa into action so he could get rid of her in a way that binds Sansa closer to him or whether he just slipped up. I've been back and forth on this issue, and I'm with you in wanting to hear other people's opinions about that.
On the question whether he's in some kind of love with Sansa or she's just a piece in his plan, I do think it's both. He desires her (in particular as Cat 2.0), but that doesn't mean that her welfare is a good in itself for him and he will use her in a way that brings him closest to his goal. As his goals include ending up with Sansa, he probably won't use her in any way that poses too great a risk to her life though.
By the way and apropos nothing, I went and tried to find what sapphires are classically meant to symbolize irl. There is little the different websites can agree on, but wikipedia mentions that they are the traditional gift for a 65th wedding anniversary. Pretty exactly 65 years before our novels start, Walder Frey married his first wife. Coincidence?
Edric Storm is widely known to be Robert's bastard, in fact Robert has even acknowledged him (not legitimized, but acknowledged, i.e. admitted publicly that Edric is his). No wonder Mel knows.
Of course, all that King's Blood talk is a bit strange anyways. Who decides who's a king and who isn't? Would Joffrey count, for example? Robb? Did Mance count, actually? I tend to think it's no more than exalted words and superstition and has nothing to do with the actual magical makeup of the world.
Addressing the claim that "sapphire = secret pertaining to this scene" is so broad as to be unfalsifiable: I think it's very important to check whether that is the case (very much so: my minor was theory of science and my heart sings every time I see someone questioning falsifiability), but on consideration and test I think it very decidedly isn't unfalsifiable.
For the simple reason that analogous theories of equal falsifiability can be falsified.
The first (and considering the conclusion only) one I tested was "bacon = secret pertaining to this scene". If you don't trust me that it's not cherry picked, I encourage you to try a random word of similar frequency yourself.
In AGOT, bacon is mentioned in several scenes:
1) Tyrion talking to the twins about Bran, his fall and his tale when he wakes up. Check.
2) Luwin reports that Bran is still unchanged. Uncheck.
3) Sansa hears that Ned went with the king to hunt aurochs and thinks that she has never seen one herself. Uncheck.
4) Rast refuses to hold back when sent against Sam. Uncheck.
5) Jon prepares to desert. At least with a broader definition, let's count it as check (cause he doesn't want to be seen and stuff).
Let's move on to ACOK:
6) Bran feasts his brother's bannermen, Luwin thinks he's done well. Uncheck.
7) Jon tells Gilly that it's extremely unlikely that they'll take her with them and it's not his decision anyways. Uncheck.
8) Jon breakfasts with the rangers. One of them boasts that he slept with three of Craster's women during the night, so I'll count that as a check even though that boast isn't very central to the scene.
9) Mormont tells Jon not to feed his bird, as he just gave it bacon, Jon sitreps. Uncheck.
10) Rangers return and sitrep (without any visible deception). Uncheck.
I'm gonna stop here, because it becomes quite obvious that even if you're very eager to find a connection of bacon to secrets, it's never going to be anywhere near as tight as the correlation between sapphires and secrets. I got 3 hits out of 10 and for the sake of argument tried to go out of my way to find hits. Maybe someone better at bendy thinking will get a slightly higher number... but nothing near the sapphire's perfect score I wager, unless "pertaining to the scene" is just replaced by "somewhere in the same book". Not sapphire grade.
Tl;dr: While it's very commendable to check whether a theory is falsifiable, this one passes that test easily. While a refinement may be possible (no idea about that), it's not necessary to make the theory meaningful - it already is.