Is the concept of The Doctor being half-human (as mentioned in the 1996 movie) ever mentioned again, and confirmed to be part of the universe's canon?
Or was this thrown away in the 2005 series, and ignored from then on?
There's a very thorough answer from Wikipedia that should answer your question:
Significant to the plot is the premise that the Doctor is half-human, "on [his] mother's side". This fact has proved extremely controversial among series' fans. The issue was referenced a number of times in the BBC books featuring the Eighth Doctor, which either seek to explain it or elaborate on it. Alien Bodies subtly suggests that it is just the Eighth Doctor who is half-human, while others books (such as Unnatural History and The Gallifrey Chronicles) suggest that the Doctor's human mother is a Victorian Lady called Penelope Gate, and his Time Lord father is called Ulysses. Another explanation is offered in The Taking of Planet 5, where it is suggested that the Doctor has become half-human as a result of repeated regenerations around humans, where he absorbed bits of their DNA. The issue was not addressed on-screen again, though in "Journey's End", a second version of the Doctor is created whose physiology had, through the unique circumstances involved, been created as a combination of the Doctor's and his human companion's physiologies; the new Doctor unenthusiastically explores his newly half-human body. In the 2008 Doctor Who comic book The Forgotten the Doctor states that, prior to regenerating, he used the Chameleon Arch to create the fiction of being half-human in order to deceive the Master.
Personally, I tend to believe that the Eighth Doctor just liked the sound of being half human. Remember the Eleventh Doctor's rules: The Doctor Lies.
This an interesting question because much about that movie has been discarded and ignored however Paul McGann is still listed as the Eight Doctor so it's not completely out of canon. I think over all though this has been ignored and can be explained away by
Rule 1: The Doctor lies
After all it has long been canon that the doctor can only regenerate 12 times however recently that was changed to unlimited. However it is still a possibility if any of the writers do pick up this story line.
Being that the 7th Season of Doctor Who shows signs of answering the question of who the Doctor is, there is a good chance that this will be definitively addressed.
Right now though the stories other than the TV Movie show the Doctor as a pure Gallifreyan.
Since season two of the reboot I always thought part of his body/DNA(?) wouldn't be fixed or predetermined. He'll get his two hearts and stuff, but the rest might be random (to some extend) or influenced by his environment (as mentioned above).
While I don't have any link handy, I remember the first episode of season two: Tennant's doctor just woke up again due to the tea fumes and the first thing he does is checking his body. It's been some time, but I think I remember him checking his teeth, two arms, hair and eye color.
This could also match River Song's "background" - just the other way around.
Here is my theory. The Doctor's DNA has a Human influence in it, but both parents were Time Lords. Perhaps, his mother, either as a punishment or for her own reasons, altered herself to be a Human for a while and unexpectedly fell in love with his father, who was another Time Lord. So, maybe his mother was essentially Human at that time. So I feel that would influence his DNA without totally making him half Human.
My evidence for my conclusion: *The Doctor's mother is hinted at as being a Time Lady before the death of the Tenth. That's what the writers said. *It has been repeatedly mentioned that the Doctor's mother was Human. *The Chameleon circuit and the Time Lords can and have made others in essence Human. *The Doctor has been punished by the Time Lords before. Maybe he wasn't the only one. *His granddaughter preferred Earth, as well. *Some Time Lords have made themselves Human and forgot(i.e. the Master). *It's mentioned that his father was this great explorer Time Lord.
If you look back at the Tom Baker era, when he was on Gallifrey he had the nickname 'Theta Sigma' at the academy. It was implied that his father had married a human woman, and so the theory of the Doctor being half-human has some substance.
Well, in the last episode of 2015 it is brought up again that he is half human. I don't see why that is bad. Makes the I protect earth at all costs make sense.