Was she REALLY the Doctor's granddaughter? Was she a Time Lord?
In the most recent new episode, "The Name of the Doctor," we see a shot (created by
Forrest Gump-style blending of old and new footage) of the First Doctor and Susan on Gallifrey, about to steal the TARDIS. That makes it definitive that she was with him from the beginning of his journey, and thus was presumably a Time Lady (not Lord, that's male) of Gallifreyan origin.
She and The Doctor obviously stayed there long enough for her to get noticed. If she were indeed a Time Lord, why would he enroll her in school on Earth?
Because she wanted it. She enjoyed the opportunity to go native and attend school. It was probably the first time since leaving home that she'd gotten to spend time with other young people (although she was probably chronologically much older than her classmates).
The thing that suggests to me that she is not really related to him and thus not likely to be a Time Lord is the fact that he sort of unceremoniously leaves her behind on a post apocalypic Earth to live out her days with an ordinary human.
I think that's misunderstanding his decision to give his granddaughter a chance for happiness with the man she's fallen in love with -- to let her leave the nest even though he'll miss her terribly. And it's hardly "unceremonious" -- few companions have ever gotten a more beautiful, touching sendoff than the Doctor's wonderful farewell speech to Susan.
One of the things that I've noticed about the early episodes is that it is not entirely clear that the Doctor is supposed to be an alien.
That's true. There were several instances when the First Doctor referred to himself as human, at least indirectly. It wasn't until the Second Doctor's era that he was explicitly established as an alien, and I don't think it was until the Third Doctor's debut that he was established as having two hearts. (There's a Hartnell episode where he's at least implicitly treated as having only a single heartbeat, though it's ambiguous.)
Which is a notion I've always found silly. What's the point of making the Doctor an alien if you are just going to "humanize" him?
One thing I liked about the movie's "half-human" retcon is that it explained why, out of all the gazillions of sentient species in the universe, the Doctor had chosen ours as his special fixation. I mean, it's pretty ethnocentric to assume that we're the only species in the universe with enough admirable qualities to capture the Doctor's affection to such a great degree. It makes sense that there should be a reason for the connection between him and us.
Although, granted, the retcon created more problems than it solved. And it did kind of domesticate the Doctor a little. Given that the intent of the pilot movie was to reintroduce the series to a new, largely American audience, maybe there were fears that said audience would be less able to identify with the lead if he were purely alien. But having him be an alien with unpredictable behaviors and allegiances helps make him interesting -- and it gives him the kind of detachment from us that lets him occasionally side with the aliens against their human oppressors.
