Have her make a horcrux by killing another human? it's pretty clear both canon and MoR that killing somebody is necessary to make a horcrux.
I don't think that's compatible with moral of any of the people that want Hermione to live.
It's quite clear that whoever introduced the troll to Hogwarts wanted Hermione killed, otherwise her broomstick wouldn't have been tampered with.
I think you're reading too much into small details.
It could very well be that McGonagall doesn't like phoenix travel, or (more likely) that Dumbledore focused on bringing Harry and Hermione into the safety of Hogwarts as quickly as possible, while McGonagall has lower risk and is also able to defend herself.
Fawkes can only transport three people at a time
Somehow that also seems unlikely to me. Phoenixes are displayed as very powerful, both in MoR and canon. Their actions are more limited by their narrow goals and maybe limits of their intelligence than by limits of their magic.
I dimly recall that in canon, Squibs are actually the children of two wizards. That contradicts Harry's finding directly.
But then Rowling probably didn't have any rules in mind about how magic inherits, so it might be impossible to come up with a good theory that explains everything we know from canon.
One thing I'm missing from this whole horcrux discussion is: What happens if you die of age, and have a horcrux?
People just seem to assume that once you have a horcrux, you won't wither and die.
But we have no indication to believe this is what actually happens. canon!Voldemort catches a rebounding killing curse, and the horcrux doesn't make him live on in perfect health. Instead he is very close to death, has no body, and needs to possess animals or other humans to extort some influence.
So what happens if you have a horcrux, and come close to dying from old age? It seems to me that your body would die, and you'd need some avenue to live again, and that is not a nice prospect at all. If you have access to a philospher's stone you wouldn't have such a problem, but then you wouldn't need a horcrux in the first place. What else can you do? Possess another human, who suffers greatly from it. Or the ritual that requires a servant of yours to sacrifice a limb; oh, and there's only a limited supplies of bones from your father, so you can't repeat it indefinitely.
In summary, it seems that a single death doesn't give you 100k+ years of live without additional major costs.
but the Hogwarts map couldn't find him when asked to find Tom Riddle, his true name.
Note that Quirrel was at the Ministry for Magic for interrogation while Dumbledore used the map to search for Riddle.
Yes. That and the fact the book is resistent to rough handling. Though of course if I were a magical archaeologist, I'd also find some spell that makes those valuable artefacts as indestructible as possible.
You mean, like, the book he gave Harry?
In canon it's definitively done.
But how?
I'm pretty sure that both canon and MoR are silent on how it's done, which is a real pity.
In canon there is a scene where Voldemort breaks into Nurmengard to ask Grindelwald where the Wand is, and then kills him. In a non-magical world I'd say that the fact that somebody can break in means that somebody can break out too, with help from the outside. Even if that's not the case in a magical world, it means that his followers could continue to communicate with him. Not good.
On the other hand there seems to be magic in canon that cannot be broken or circumvented, except for a very specific trigger. Think of the Fidelius charm, which hides a building from everybody, except those that the secret keeper has told the location. Or the potion in the cave that must be drunk, and cannot be vanished, transigured or otherwise "magiced". Maybe a similar kind of "absolute" magic exists that can be used to imprison people reliably. So reliably that no Auror need to stand guard, and are tortured with humming.
Did you notice that from Quirrel's perspective, that's exactly what he has done to/with Harry? Killing Hermione had the effect of hardening Harry's resolve, and removing some of his scrupels. For Quirrel that's "stronger".