Yes, I do miss the old LJ community. I also to recognize that it's not only changing technology that is to blame; it's also that we're "old" and fandom has gone elsewhere, alas. :)

I dismiss the timeless child from my head canon. I don't have any precise backstory for the Doctor's family of origin, but I generally assume he came from a normal, fairly upscale Gallifreyan family, as Time Lords, in general, seem to be an upper class on Gallifrey.

Most of my mental ficcing has been around how the Doctor ended up with Susan, and my general story is that he originally fled in the TARDIS with his daughter, who eventually married a Englishman (hence Susan's apparently "real"/only human name and the Doctor's weird tie to England). Long story short, both Susan's parents died, leaving her with the Doctor.

I see "Susan's Grandmother" is the Doctor's ex; he seems like the sort of person who would not do well in a LTR, and I don't understand why we have to default to the cliché that she's his tragically dead true love. So she'd be back on Gallifrey for all this, and they'd have a custody battle over Susan, which he wins by refusing to comply with the inevitable order to return her to Gallifrey. I think, though, that Susan eventually made it to Gallifrey and maybe had quite a good relationship with her grandmother. Until otherwise shown, I assume they both died in the Time War.