I'm gonna state up front that my knowledge of canon is limited to the aired episodes. I know bugger-all about fanon sources such as the novels or audio adventures. So this is based on my (somewhat limited by time and fuzzy recollection) knowledge of the series alone (which includes most of Three, all of Four, most of Five, parts of Six, vague rumours about Seven, and Eight on video tape somewhere in my basement).
Gallifrey go BOOM (and should stay went BOOM)
What defines the new series for me—and defines the Ninth Doctor in my mind—is the destruction of Gallifrey.
The Time War is being set up as the arc of the season, and I have become fascinated with it, because it has significantly changed both the character of the Doctor, and the Doctor Who universe as we know it.
The way I always understood it, the Doctor stole a TARDIS because he was disgusted with his people, and the fact that they had the power to help, but not the compassion to use that power. He held Gallifreyan society in contempt, and rebelled.
With the destruction of Gallifrey, the Doctor's status changes. His entire sense of self is changed. He's not a rebel Time Lord any longer. He's now the last Time Lord. And that changes the playing field considerably. It changes how he views himself, it changes how he views his people, and it affects any relationships he may have with anyone he meets.
Even in exile, the possibility still existed that Gallifrey could change. That the Doctor could be a force of change for his people. That he could have a home to go back to that was not the home he left—that he chose to leave, and that he was barred from returning to. That possibility meant that there was hope. Even if it was a hope he wasn't aware he held.
But are you still a rebel, if the world and the attitudes you were rebelling against no longer exist? And what do you do, when you lose something that is such a huge part of your sense of self? How do you adjust when a hope you may not have realised you held out, is extinguished completely?
I said, after "End of the World" that the loss of Gallifrey "woobified" the Doctor. I only half-meant it as a joke. The way, in the 5 aired episodes, the Doctor has been affected by the destruction of Gallifrey has made him more vulnerable than we have seen him in previous incarnations.
His anguish over the Time War was hinted at in his exchange with the Nestene Consciousness. "I couldn't save your world—I couldn't save any of them," was a line read with considerable emotion, as the Nestene blamed the Time Lord, and his reaction was emotional. While some incarnations of the past wouldn't have been as hesitatant to use the anti-plastic the moment the threat was identified, this incarnation held out hope that the situation could be resolved without destroying the Consciousness. Despite his experiences with the Autons in the past.
His reaction to Rose's quite reasonable question of where he was from out of proportion to her query. His response is to practically shout "This is who I am. Right here, right now. All right? All that counts is here and now. And this is me." As a viewer, I interpreted his response as the Doctor clinging to the present, because the past is gone.
The Gelth used the Doctor's guilt over the Time War to manipulate him. It's strongly implied in "Unquiet Dead" that they used Gwynneth's gift to "read" him, and play on his sympathies to get what they wanted. They played him, pure and simple, and Earth almost paid the price.
We don't get an explicit reference to the Time War in "Aliens of London", but what we do get is an examination of the Doctor's compassion. And you may laugh at the "Pigs in Space" reference. But the Doctor's empathy for the frightened animal's death is an example for the audience of the Doctor's "humanity." Even thought the character is set up to be "alien", it's his very alienness that allows the character to hold up a distorted mirror to reveal what is best in humanity. In the moment where he bends down to offer comfort to a dying animal, we see a very pure example of compassion. It's that same compassion that he saw as lacking in his own people.
This Doctor is alone. Before, he chose it. He chose to leave Gallifrey. He chose to rebel. He chose to continue travelling. Now that choice has been revoked. He is alone in a way he never has been before. Which brings me to his choice to take on a new companion, and how his relationship with Rose is defined by his loss of Gallifrey.
The Doctor has witnessed—and been scarred by—an ending. So it makes sense that he would look for a beginning. By viewing the Universe through Rose's eyes, he can experience it as something new and wondrous, seeing possibilities for the first time—instead of seeing his piece of it that is no longer there. He takes delight in her delight at things that are old hat to him, but an extraordinary new adventure for her. It allows him to experience the universe in a way that might otherwise have been lost to him, because he has found a companion with a sense of adventure and wonder equal to his own.
It's established in "Rose" that Rose is in many ways an adult among children. Jackie turns Rose's ordeal into something all about her, including making disparaging comments about how the ordeal has "aged her—skin like an old Bible." Mickey is concerned only to the degree of missing part of a football match. Rose accepts this from the two people closest to her, and when both of them are threatened, Rose is the one who steps in to protect them.
When the Doctor tells Rose to "leg it" because the Nestene Consciousness is beginning to transmit the signal, Rose's first instinct is to call her mum and warn her. Not self-preservation—but looking after her loved ones. While Mickey is utterly useless, Rose stays by his side. When the Doctor is threatened, Rose takes action to help him, despite the potential consequences to herself. Rose initially rejects the Doctor's offer because "someone needs to look after [Mickey]." It's only when she learns that the TARDIS travels in time as well as space that Rose allows herself to make a choice to actually benefit her—instead of catering to the needs of the people around her.
While the Doctor has had travelling companions in the past, I don't think—emotionally—he has previously needed one as much as he needs Rose, in the wake of the destruction of Gallifrey. And that fundamentally changes their relationship, in comparison with companions of yore. It makes him vulnerable—needing someone always makes you vulnerable—but it also make him stronger, because he's forging a strong relationship out of that need that is more of a partnership than many of his previous liaisons with companions.
And because he is vulnerable, he's not above using emotional blackmail. In "World War Three" when Jackie invites him to tea, he dangles the wonders of the Universe like a carrot from a stick, literally, to force Rose to choose between him and her family. As if he's testing her conviction. He wants and needs her that badly. He is intensely uncomfortable with families (there are repeated variations of the line "I don't do families" over the course of the two part episode) and appears to actually flee Rose's farewell with Jackie. Yet he began his journey in the TARDIS with his granddaughter in tow. So it's worth mentioning that he is so visibly uncomfortable with Rose's ties to her family.
By choosing Rose, the Doctor has involved himself with someone who will challenge him the same way he tried to challenge his people. Rose gives as good as she gets, has strong opinions, and doesn't allowed herself to be cowed by him. This is particularly apparent in "The Unquiet Dead" when Rose argues with him regarding using Gwynneth to assuage his own sense of guilt. Rose may have been right for the wrong reasons—but she was still willing to stand up for herself, and stand up for someone whom she saw as her responsibility, whom she did not believe capable of standing up for herself. This pattern of behaviour echoes the Doctor's own desire to use the power he has as a Time Lord to help people, rather than merely observing and recording. But also to actually experience life, instead of merely witnessing it. It's that thirst for adventure that defines both the Doctor and Rose, in that respect.
Unlike many previous companions, Rose was not stranded. She is not merely a hitch-hiker. She also made a conscious choice, much as the Doctor did when he left Gallifrey. She wasn't content with the life she had, was frustrated by the complacency of the people closest to her (as illustrated by Jackie's comment that working in Henriks was giving Rose "airs and graces" and Rose's dismay over lunch with Plastic!Mickey over having limited options for employment due to her lack of education). So the series has consciously set Rose up as a much-smaller-in-scale mirror of the Doctor, in many ways.
Ignoring the meta-question of whether the pacing of the growth relationship has validity, or is being rushed, the way it is presented in canon is this: Rose looks after people. And the Doctor needs looking after. That's how their relationship starts. And I don't think it would have started—or grown so close so quickly—if it hadn't been for the destruction of Gallifrey.
As a viewer, I am clinging to the idea that Gallifrey is gone forever and cannot be visited or resurrected, and the Doctor is the Last Time Lord, because it gives the story a dimension is has not had before. And that is a defining aspect of this new series which I would like to see explored.
To me, it's like Superman.
(What? We're in Smallville, now? No. Trust me. This relates.)
The heart of the character of Superman is that Krypton is gone, and he is the last of his kind, and he is alone. It gave a sense of epic tragedy to a silly little four-colour comic aimed at 8 year olds and GIs. It's part of why the character has endured for 60+ years, because it took a guy who was wearing his underwear on the outside, and gave the story a sense of pathos.
However, as time went on, that core was eroded by the appearance of a cousin (Supergirl), a shrunken city (the bottled city of Kandor), and the villains in the Phantom Zone. You chip away at an epic story, until it's a joke. He's the Last Son of Krypton (Except For All These Other Folks, and the Dog, Horse, Cat, and Monkey*).
If the writers of the current series are going to destroy Gallifrey—wipe it and all other Time Lords out of existence, possibly even erasing their very existence—then the impact of that decision is diminished if you have the Doctor discovering other surviving Time Lords, or be able to travel into Gallifrey's past. It would actively undermine the pathos for me. It would feel, in short, like a cheat.
The whole dynamic of the central relationship in the series—the Doctor and Rose—is delicately balanced on the Doctor's grief over the loss of Gallifrey. I'm not saying that it's impossible to resurrect Gallifrey or the Time Lords and still make the series captivate me the way it has.
I am saying there would have to be a serious amount of fancy footwork to make it not feel like the whole Time War arc was nothing but a put-on. A smoke and mirrors light show. Because from the moment Jabe first accessed the Metal-Mind on Platform One and scanned the Doctor, I went as a viewer from "Wheeeee! My show is back and it's so much fun!" to "OMG I am now thoroughly obsessed."
The destruction of Gallifrey changes everything, and I am fascinated with how those changes make themselves felt with each episode that is transmitted. I'm enjoying learning the new terrain, now that the face of the playing field has been so dramatically changed.
I just don't want, at the end of episode 13, to have the lights come up and find out all that depth was just shadow-play.
*There were, I am so not lying here, a Kryptonian wonder-dog, super horse, super cat, and super monkey named, if memory servers, Krypto, Streaky, Comet, and Beppo. Okay, I could not MAKE THIS SHIT UP.