If I remember correctly, in the episode of the original series when Ikari Gendou and Fuyutsuki go to Antarctica, the sea was also red. And, as we see in the Rebuild, Angels bleed copious amounts of blood. Now imagine how much would be spilt if you'd reduce the Giant of Light to it's embryonal form: quite a lot I imagine.
And in an interview about the Rebuild, Anno stated that he wanted "old hard-core fans and even fans who just know Eva from pachinko to view it as a single (i.e. stand-alone) movie." Newtype Japan 2006
http://www.evaotaku.com/html/neweva.html
And I think that's just Rei flying towards the screen just before Shinji wakes up for the first time in the hospital. It might be a residual effect of piloting Unit 01 since...
Now, we've still got a few problems:
One: Adam is number 1, Lilith is number 2, Sachiel is number 4. Number 3 is absent, though it could be Moon Angel, which would explain the red splash on the moon (note that the Moon Angel is, like Lilith, missing a few limbs).
Two: Misato calls the Angel in the basement "Lilith". I checked and in the series she mistakes it for Adam. Are we in for another mix-up or... Could the Giant of Light (the cause for Second Impact) NOT be Adam?
Three: If you look closely at "Lilith's" mask, you can see it's different from the one in the series. If you look at Moon Angel, you can see the mask from the series on it's face.
Now I've got a different theory:
The Giant of Light is not Adam. Adam is on the moon and Seele is using it's progeny to fulfill the Dead Sea Scrolls.
Kaoru has been bumped up to number 6. If you look closely at the scene when Kaoru wakes up, you see 4 caskets that've already been opened. Which would account for the Angels we've already seen (Lilith, etc).
Now who's the Giant of Light, the one that caused Second Impact? It could be Unit 01. In the series we've seen how important Unit 01 was for Seele's plans.
There are still some flaws in this theory: How would Kaoru already know Shinji? Why does Kaoru say "The Third again, huh?" when he's referring to Shinji?
Of course, there's always the craziest possibility of all:
Shinji is the Third Angel, and the Third Children, making him a third again.