I think Peter’s a brilliant actor (the best conversation I ever had with an actor was a phone call with Peter about Islington, and about how Islington always believes itself to be in the right). I can’t wait to see what he does. I’m glad that we’re getting an older Doctor — we’ve had two puppies, it’s time to see someone older. It’ll change the nature of the relationship with Clara in interesting ways, for a start. ... [I'm] just excited to see what kind of Doctor he makes. He’s an Academy- Award winning director, an amazing actor and I really liked him when I worked with him before.

On introducing a female Doctor:

Do I think it’s time to cast a woman as the Doctor? Not yet. Not quite. And lord, if and when they ever do that, I want them to keep it the biggest secret in the world until we see it happen on our screens during the regeneration.

If I were show-running (I’m not) I wouldn’t cast a woman as the Doctor yet, and it would absolutely be on my list of things to do in the following regeneration. (I was the one who wrote the line about the Corsair changing gender on regeneration, in “The Doctor’s Wife" after all, and made it canon that Time Lords can absolutely change gender when they regenerate.)

Some of that is stuff I’d find hard to articulate, mostly having to do with what kind of Doctor you follow Matt Smith’s Doctor with: someone harder and much older and more dangerous and, yes, male feels right to me, as a storyteller. Where you go after that, ah, that’s a whole new game…

On there ever being a black Doctor:

Would I like a person of colour as the Doctor? Absolutely. Paterson Joseph was the Marquis de Carabas in Neverwhere, because he aced the auditions, and beat all the other actors, mostly white, who tried out for the role. I’d want that kind of performance at the audition for the Doctor. And there are certainly actors good enough out there that it feels like a missed opportunity. ... And yes, I have no doubt there will be. (I know one black actor who was already offered the part of the Doctor, and who turned it down.) Just as there will be a female Doctor.

i09  speculates that this last comment means that a black actor was offered the role of the twelfth Doctor and had turned it down before it was offered to Capaldi.  Personally I'm not convinced that's what Gaiman means, not least because that would imply he has an awful lot more inside information on the current status of Doctor Who than we thought, but also because that's simply not what he says.  Then again,  perhaps that is exactly what he means by it.  A comment like that is bound to raise more questions than it answers, however, and so I suspect we'll hear the truth of it before long.

And finally, on the topic of just how the show might get past the thirteen regenerations limit:

Well, that was the rule when we had Time Lords running the show. (And they were the ones who gave the Master his extra regenerations.) There aren’t any Time Lords any more… 

MY OPINION (which is not Canon) is that the regeneration limit is a lot like the speed limit. You can break it, but things get a lot more dangerous if you do. The Time Lords were the traffic cops: they enforced the limit. With them gone, the Doctor can keep regenerating beyond 13, but with consequences.

 Not canon yet, Neil.  You can always write some more and make it so!