But isn't the "real" Doctor also supposed to be half-human (as per the TV-Movie)?
That was alleged in the movie, but everything since has ignored it. Indeed, "Journey's End" seems to have been written specifically to refute it, because it repeatedly establishes the idea of a half-human Doctor (the Metacrisis clone) as something new and different.
It's worth remembering that in the movie, it's unclear whether the Doctor is serious when he says he's half-human. He's about to tell Grace something in the strictest confidence, something very secret, and then a stranger butts in and asks about their conversation, and the Doctor tells
him "I'm half-human on my mother's side." So it doesn't seem that can be the real secret. The Master also came to believe the Doctor was half-human due to his retinal structure, but there could've been something else going on there.
Unless the new cycle begins with a "reset" and has 12 more lives after it. That hasn't been made clear yet.
It hasn't been made clear, but I didn't hear anything to suggest that it's true. The only thing I can think of is the swansong to Matt Smith, but that had to do with the new lives taking longer than normal to kick in, according to the dialog, not with going from life 1 to life 2.
No, that's not at all what I'm saying. Think about it. The Doctor had one birth
followed by twelve regenerations, hence thirteen distinct lives. But if you add a new cycle of regenerations, then there's no "second birth" added along with it, because the Doctor's already a grown man. A regeneration cycle is twelve regenerations, and the initial birth is an extra. Or rather, the first life is what a Time Lord gets
before the regeneration cycle engages at all, and the 12 regenerations are the extras. So if he gets a new regeneration cycle, it wouldn't be 13 + 13; it would be 1 + 12 + 12. And the first of those 12 regenerations would have to be used up turning Smith into Capaldi, just as the first of his previous regenerations turned Hartnell into Troughton. And that would leave only 11 Doctors after Capaldi, at least until the cycle is reset again. So no, I am absolutely not saying that one regeneration was used up to turn Smith young again and then a second was used to turn him into Capaldi. I'm saying that turning him into Capaldi expended the first of the 12 new regens, leaving only 11 (until the next reset, of course).
The alternative is that the "reset" somehow gave him a totally fresh start that was akin to a second birth, another "starter" life in addition to the 12 do-overs. But that's harder to justify, and I don't think it was the intent.