In the first season of Doctor Who there is a companion named Susan Foreman played by Carole Ann Ford. She always refers to the Doctor as 'Grandfather'. Has it been established that she actually is the Doctor's granddaughter? This would obviously mean she is Gallifreyan or a Time Lord. IIRC, even that is ambiguous.
The Ninth and particularly the Tenth Doctors don't mention Susan by name, but do reference fatherhood on several occasions (the Ninth Doctor says that his entire family died in Father's Day, and that he "knows the feeling" of having been a father and grandfather in The Empty Child; the Tenth says he was/had been a father in both Fear Her and The Doctor's Daughter).
Although it is not explicitly stated anywhere, the natural assumption is that Susan is/was his natural granddaughter and therefore Gallifreyan. It is unlikely she is a Time Lady since she was travelling with the Doctor at age 15 and then retired into a quiet life with David Campbell, although since Time Lords were taken to seen the vortex at age 8, it's possible she did this prior to starting to travel with the Docter.
When Susan appears in the "Death Zone" in The Five Doctors, she appears to be reasonably familiar with Gallifrey, although it's possible that she picked this up from travelling with the Doctor rather than from being a native.
Susan also appears in the 8th Doctor's audio adventures. Her and her son Alex, who doesn't meet the Doctor until he is about 17. The Doctor runs DNA on Alex and finds he is only 7% Galifreyian, but this doesn't make sense to him (he should be more, according to the Doctor). He doesn't really delve into this though. When Alex finds out the Doctor has 2 hearts he asks his mother, Susan, if she does as well. She doesn't really answer though.
Alex also wonders to his mother what to call the Doctor, "Great-grandfather" she suggests. He ends up just calling him "Granddad." The Doctor also invites Alex to travel with him - and planned on leaving his TARDIS to him - but Alex rejects his invitation and instead travels with Lucie (the 8th Doctor's old companion) on Earth.
Before that - when he first met the Doctor (6 months prior) - The Doctor wanted Alex to have an education on Gallifrey so that he could fulfil his full academic potential as a Time Lord. However, Alex saw Earth as his home and decided to stay with his mother.
It should also be noted that Carole Ann Ford reprises her role in the Audio Drama's. Paul McGann's son, Jake, plays Alex.
Alex is killed in the Dalek re-invasion in the Audio Big Finish Drama "Lucie Miller / To the Death"
IMO, Susan is the Doctor's biological granddaughter. He refers to her many times as his granddaughter or that he did have one - constantly as the 1st Doctor and multiple episodes after the restart, particularly The Empty Child and The Rings of Akhaten. Presuming only the TV show and movie (there are numerous inconsistencies in other media), there is no evidence whatsoever that she is otherwise.
As far as being fully a Time Lady, there are two points that have to be separated: did she come from Gallifrey, and is she a Time Lady?
Susan shows every indication of being from Gallifrey. She has knowledge consistent with that idea, even going so far as to say that you could not do math in three dimensions, you had to use all five (Time and Space being four and five). The Doctor states that they were both exiled from their home but would return eventually. This implies strongly that Susan also came from Gallifrey (An Unearthly Child). Also, Susan was with the Doctor when he left Gallifrey in the TARDIS (The Name of the Doctor).
Lastly, during the movie, the Doctor states that he is half-human from his mother's side, the only time that he says this. If this were true, I'd find it puzzling why other Time Lords, such as Romana and the Master, did not mention anything about the Doctor being half-human during the series. Also, during The End of Time, the Time Lords appeared to accept the Doctor's mother as a complete member of their society, which is unlikely if she were human.
One possibility is that he remembered incorrectly due to his run-in with amnesia, which The Doctor experienced due to human anesthetic affecting his recent regeneration. Another is that he simply thought it was fun to lie; as River Song says, Rule #1: The Doctor lies. Or, he's telling the truth. In the first two cases, Susan is completely a Time Lady. In the third, she's at least 1/4 human and 1/4 Time Lord. The rest of her genealogy would depend on whether the Doctor's child married a pure human, pure Time Lord, or a mix.
This would obviously mean she is Gallifreyan or a Time Lord.
As it has been stated in many episodes, not all Gallifreians are Time Lords. Time Lords are a ruling class, that children of Gallifrey must be initiated into. While Time Lord and Gallifreyan are used interchangeably alot, the doctor makes the differentiation often.
The latest proof was nu-who season 8 episode listen. The Doctor, pre-initiation, was crying in bed. His parents (or school attendants?) Talked about the options. It was a choice to become a time lord.
So to the question, nothing has indicated that Susan Foreman is a Time Lady, and her status as a full blooded Gallifreyan is also questionable as we don't know the pedigree of Susan's grandmother, father or mother. But one thing that has never been put into question is that the doctor has always called her his granddaughter.
Something I noticed recently is that the founder or head of a Gallifreyan House is titled "grandfather" now im not saying that it means anything, but when you add that Susan simply "recognised" the Doctor as her grandfather along with the alluded-to link between the Doctor and the Other. Not to mention in recent years his marriage to River, the creation of Jenny.... The Doctor's affinity towards earth (while convenient to the story, obviously) is sort of in question a bit, but (and I'm not saying it means anything) but what if the human race is detined to become the Time Lords? Okay, so its something that should never actually be said on the show, but it's a little idea that's been rattling in my head
I believe she is his actual biological granddaughter. She is the only companion to call The Doctor her grandfather. However I don't believe River Song could be her mother. Remember that River and The Doctor have personal timelines that work in reverse of each other. River meets the 12th and 11th reincarnations but her life ends during that of the Tenth Doctor (David Tennant) so she would never have met the first incarnation (William Hartnell). Hence The Doctor has been married at least twice and Susan is the granddaughter of some other earlier wife of his.
I'm also reluctant to believe that beings of two different species can reproduce. This goes against one of the fundamental laws of biology. A human and a chimpanzee can't reproduce even though they share very common DNA. Hence a Gallifreyan and a human should not be able to reproduce. I know this occurs a lot in Sci-Fi (esp. Star Trek) but goes against biology. Hence Susan Foreman must be 100% Gallifreyan.
In the newest season, Peter Capaldi's doctor states that he once convinced the Master he was half human, indicating the 8th doctor's movie. If the statement by Capaldi's doctor is true (which I believe it is), then he is 100% TimeLord/Gallifreyan, and Susan (assuming she is his biological granddaughter and depending on his spouse and his children's spouse) would be anywhere from 25-100%
For what it's worth, I believe she is, otherwise the doctor's "lost" family would still have to be identified at some point anyway.
Of course it's possible that Susan is River Song's daughter with one of the Doctors, but how Moffatt would explain how the first doctor didn't recall River ... Oh! One Doctor had his memory partially wiped by the time lords, didn't he?
As for why Earth is so special in the universe: aside from being the planet where the series is filmed.
Ok, remember the episode set 5 billion years in the future on a space station orbiting near Earth? Humans had genetically engineered many creatures, including themselves, and sent them all over the Galaxy (and beyond?).
The Daleks were heavily genetically modified Khaleds, humanoid relatives of the Thraals who had almost exterminated them in a nuclear war. This and Davros's mad scientist syndrome go a long way to explaining their psychopathic paranoia.
Time lords look suspiciously humanoid too. So I suggest that over 5 billion years in the future, genetically modified humans are sent to the planet Gallifrey. Gallifrey itself time travels, notice that?
I further suggest that many species of aliens we repeatedly encounter were settled on their worlds in the distant past, by primitive time-travelling post-humans: our own descendants.
The Time Lords merely rediscovered skills their ancestors had used to plant them on Gallifrey. Either that, or perhaps they are the post-humans who did all the scattering of species. Much about the first Rasillon and his colleagues is not known, but he certainly does have a "the Universe is mine by right to do with as I please" syndrome.
So, many of the aliens are descendants of humans. Therefore, if they were to permanently conquer 20th or 21st century Earth, they would destroy all their own timelines. So it doesn't happen. The reason they even try is that their own origins are lost in the remote depths of time immemorial.
I think that in "the end of time part 2" the one time lady that helped Wilfred and showed her face to the doctor was actually supposed to be Susan because it never really explained what happened to her and they used her to show everything the doctor was giving up to defeat the timelords in addition to giving up his beautiful home planet.