Blood-C/WMG

Saya's Eye will never heal, is the price of a wish: to become capable of killing humans

 * Jossed, by the last shot in the movie trailer after episode 12. Unless we're looking at a reflection, for some reason.

Saya is Saya from Blood+ under a different last name.
This is hinted at by the brief flashes of what looks like her fighting Chiropterans.

The teacher is an Elder Bairn
She did know the story, or at least one ridiculously close.

The Twins didn't really come Back From the Dead
Kinda obvious.

Tokizane knows about Saya and the Elder Bairns
This is why he is so protective of her.
 * He does, she told him on episode 8, then he died on episode 10.

Elder Bairns and humans come from the same source
Their name means Elder Children, which may mean that they are humanity's "older sibling"
 * In episode 6

Saya is a Vampire which is why her eyes glow red
This is also hinted at when we see increasing amounts of shots of her blood cells.

The dog is an Elder Bairn
He does seem to watch over Saya in at least one of her battles, and causes a reaction in her.

Saya's mother was an Elder Bairn
She fought the Elder Bairn and her daughter is definitely not normal.

Saya has amnesia
Pretty simple, we see her trying to catch a light in her mind, that later looks like a sword and is accompanied by flashbacks to someone fighting Elder Bairns.

Saya's mother didn't fight Elder Bairns
It was just Saya before she lost her memories.
 * Episode 6 suggests that her memories weren't lost, so much as removed or hidden by her father. He managed to put her to sleep with a simple hand gesture.

Saya's dad is manipulating her to a evil purpose.
He does love her, but he believes it is for the greater good.

Fumito-san behind the whole plot.
Or... he's on the bad guys's side.

Saya isn't actually her father's daughter.
She is being raised as a weapon, she was found similarly to Blood+!Saya, she's her dad's Replacement Goldfish for his wife.

Yuka is Saya's sister queen.
A scene in the OP makes her seem to know more than she lets on.

Yuka's on the bad guys's side.
Like the above WMG.

Saya's teacher is researching her.
A lá Dr. Julia from Blood+, she said that, in that school, she had a chance to research something she wouldn't be able to see anywhere else, all this while talking to Saya.

Tokizane has another form - the dog.
That's why he is also constantly fatigued and late to class.
 * That would be impossible, since in episode 4, they would be in the same scene, it can also be Fumito or the Teacher though...
 * Actually, they're not. If you look closely the dog disappears right before Saya falls from the tree, followed by Tokizane cushioning her fall at the bottom of the tree. A stretch because of location, yes, but doesn't rule him out...

If the singing starts again, run!
No, not for the obvious reasons. Saya is eventually going to lose control. It's pretty much guaranteed. And, when that happens, she will become a happy cheerful schoolgirl... who likes to kill people. And she WILL sing her happy song.

Saya will be forced to kill all her friends.
They will all slowly become monsters, until she has to kill them.

The dog is actually

 * A part of Saya's subconscious trying to warn her.
 * A fox.
 * Saya's mother.
 * The Dimension Witch.
 * Or, more likely, Watanuki.
 * A Witch; take your pick of which.

The show is a direct continuation of Blood: The Last Vampire
As Saya's visions become more and more clear, we see images of a train car getting splattered with blood, along with flashes of what is clearly a chiropteran. I posit that these are flashbacks from the opening scene of Blood: The Last Vampire.

The Elder Bairns feed on fear.
Why is it that survived the attack on the school? Because, unlike everyone else,.

The Guimave Saya eats had blood in it
Fumito mentions that its color is based on whats in it, and Saya seems to like them an awful lot....
 * The coffee is rather suspect too. I don't think Saya's father drinks different coffee from hers because it's decaf. I think his is deblood.

Saya's father is a vampire/ elder bairn.
The cuts on Fumito's arm is from being fed on
 * The Elder bairn also seem to be on a First-Name Basis with him and haven't tried attacking him when he's alone so far. The one in ep 10 completely ignored him standing 3 feet away but immediately went after Tokizane.

Blood-C is a prequel to Claymore

 * Fumito's experiments with do not end with this story. Ultimately, more half-human-half-monsters will be created with the large collection that he's hidden away and the most successful ones will become the first Claymores.