I always figured that the half-human thing was a joke of sorts, the Doctor referring to how human in behaviour he has come and how Earth has pretty much become his new home (He does give it extra speshal attention).
I think someone tried to retcon it by saying the blood transfusion in the San Francisco hospital messed up his biology, so the eighth Doctor actually was half human, but the rest weren't.
It's mainly, I believe, the work of Andrew Cartmel and Marc Platt. Cartmel of course came up with the whole "More than a time lord" concept, and Marc Platt wrote up most of the Time Lord's background. In addition, Platt and others (in the book) established the concept of "Looms" that established that all time lords were basically born of test tubes or genetic databanks or something like that, and didn't have traditional families (They were basically sterile). In order to explain Susan they said she was in fact the 'others' daughter and that she recognized the Doctor as the 'new' other. Not that it should be held against the NAs, necesarilly. While the loom/other concept is no longer 'canon', there's still plenty of good books in the line, and many of the writers from them write for the new series. Also of course Human Nature was adapted from the novel.
Ahistory. Cold Fusion for his children. And his wife. Whom he's still mourning come The Infinity Doctors That one actually doesn't matter. He's on the run from the Time Lords, yes, but they put him on trial in "The War Games" not for running away. The Dark Path. Divided Loyalties. The Leekley Bible. Does anyone care? Funny. I thought the Doctor's joie de vivre made the character work. I thought the Doctor's thirst for adventure made the character work. I don't watch Doctor Who because of the mystery. I watch Doctor Who because of the adventure.
To some extent, I almost think that mystery is overrated. For example, one of the things that frustrates me about Torchwood is that they act like Captain Jack is such a mysterious figure when, in fact, anyone who has watched his Doctor Who appearances knows far more about his past than his own Torchwood team. Although, perhaps it's best for these questions to be answered in the novels. That way, the stubbornest of TV fans can ignore them at their leisure while those who are interested in knowing can know. As to the Doctor's name, I believe that "Doctor" is his name. (Perhaps, in its full form, it's something like Doktoraknikosilan.) Well, disguising a TARDIS would be a bit of a problematic issue anyway. A police box is an oddly perfect shape because it has doors that a normal person can fit through. Perhaps, with a working chameleon circuit, when the TARDIS first lands, it scans the immediate viscinity for the nearest object that is roughly TARDIS shaped and copies it, not necessarily taking context into consideration.
"'Spock?'" "Well, don't you ever get tired of just 'Doctor?' Doctor who?" - The Doctor and Rose, "The Doctor Dances" " 'Doctor.' Doctor who? It's more than just a secret, isn't it?" "What did you see?" - Reinette and the Doctor, "The Girl in the Fireplace" "There is no name. Fascinating. Why would a man hide his title in such despair?" - Lilith, "The Shakespeare Code" "I like it when you use my name." "You chose it. Psychologist's field day." "As you chose yours. 'The man who makes people better.' How sanctimonious is that?" - The Master and the Doctor, "The Sound of Drums" "River, you know my name. You whispered my name in my ear. There’s only one reason I would ever tell anyone my name. There’s only one time I could…" - The Doctor, "Forest of the Dead" Seems to me that they've been dropping hints -- especially in Steven Moffat scripts -- about the Doctor's name from the very start
the thing that makes Doctor Who Doctor Who to me is: a bloke flying around time and space in a blue phone box, often accompanied by some fit totty, fighting a buncha aliens and who regenerates when he's mortally wounded. anything else is window dressing. or bollocks.
As for the police box, who is to say he didn't first land on a street corner, the Tardis diguised itself as a police box, then the circuit broke when he moved it to the junkyard and he just didn't realize it was broken until he ended up in 50,000 B.C. He probably didn't pay it any mind in the junkyard because maybe the circuit only kicks in when a time shift is indicated, not a location shift. Take that cannon or canon, or whatever!
the phone box, I get that it might not attract alot of attention in modern day earth, but on alien planets? would we not all be interested if an alien phone box appeared on our street?
The Doctor and Susan visited Earth prior to An Unearthly Child-- a couple episodes mention their previous outing to the French Revolution. Just Wars establishes that it was the Doctor's first ever visit to Earth, and that he was impressed by seeing the toppling of the old order.