In my head canon changing genders is a choice, as statistically speaking 13 heads in a row in a system that should be heads or tails is, er... difficult to explain. We know that voluntary regenerations are more controlled, so I like to think that in a voluntary regeneration if a Time Lord wanted to be a woman - like the Corsair - they could make that change, influence it. Like a human choosing to have their gender reassigned. The Doctor is just dying in a less controlled manner each time, so he's defaulting to what he was born as each time. That's how I see it until the show proves to me otherwise.
I personally do sort of view the Doctor fundamentally as a male character, though, and I think the show would have to rejig his personality more forcefully than any other regeneration to sell that if he changed gender. Ethnicities - not so much. Bring on a black Doctor or an asian Doctor or whatever. There's just certain traits about the character that seem particularly masculine. They could be wiped away by regeneration, though. I wouldn't be mega upset if it happened, I'd embrace it.
I do agree that colour-blind casting can be dangerous, but I also agree with something Moffat said - If you set out to cast a white man, a woman, or a black man specifically, you've sort of already failed as they may not be the best person of the entire pool available to you. I do think Doctor Who has a duty to provide the young girls who watch it with positive female role models, but that doesn't necessarily need to be the Doctor. Moffat's era hasn't been great at that, but hopefully that improves.
PS: Know who'd make a great Doctor? Adrian Lester. He'd be fucking AMAZING. God. Incredible.
http://www.youtube.com/watch?v=EFnkSGH3LL8 As of right now, he'd be my dream 13th.