The thing that screamed out to me on first viewing continues to scream out, and I register sustained surprise that fanfic hasn't dealt with it: The Doctor as serial cradle-robber.
Now, that's a jokey and insulting way to put it. But the more serious phrasing is: He's a 900-year-old Time Lord with vast knowledge and experience. What about a 19-year-old uneducated Briton appeals to him? What drives him to seek that relationship? What does he get from it that he's lacking? This is a serious engine of story that drove some of what's in the season I watched. (Haven't seen enough of the first 10th Doctor run to know what it deals with.)
There are a number of story engines from Rose's point of view as well. One repeated theme in the 9th Doctor's episodes is Jackie's insistence that the Doctor protect Rose, and the Doctor's adoption of that as an imperative. What's Rose going to make of that? She's 19, which is well into the teen pushing for independence stage; past it for many people. She's crushing on the Doctor. Tempting to let the Doctor take care of her, but Rose probably won't yield. (If she does, I yawn and move on.) She's going to want to prove herself. That drive vs the Doctor's over-protectiveness == story engine.
Can't write any of this myself. Not yet, anyway. I'd need to see considerably more of the Who canon before I could plausibly attempt writing 9. Writing Rose is another set of challenges. She's culturally alien to me, but culturally close enough to a large segment of the reading audience that the plausibility hurdle is high.
Don't know what the stories are with the 10th Doctor and Rose. I think if I wrote anything about them right now, it would be anti-shippy: what Rose would have to figure out is that relationships with the Doctor aren't healthy for short-lived humans. Realize that it's an adulation for see-the-universe-thrills exchange, and not much more. A fling with #9 and Jack at the same time is one thing; please enjoy. Being smothered by needy #10 is another.
But, er, I need to have watched more than 4 eps of 10 before I decide I really have a clue.