The community memories are here, sorted into semi-useful categories. But here's some of the things that've come up most often, and a few that get asked a lot all over the place in general.
1. Sleep and the single Time Lord
Short answer: The Doctor does sleep at least occasionally but he also goes on a bit about humans sleeping all the time. Read into that what you will. (This one gets asked an awful lot. I suspect it's for pornography...)
2. How old is the Doctor?
Short answer: Umm, we're not sure. The Doctor gives out roughly sequential numbers until he's suddenly quoting nine hundred again in Aliens of London, a figure that's a good century smaller than the last number he came out with.
3. What's the Doctor's real name?
Short answer: We don't know.
Slightly longer answer: Everyone on Gallifrey refers to him as the Doctor, and that's the name he's always introduced himself with. Way back in An Unearthly Child he wasn't called Doctor Foreman. WOTAN in The War Machines actually called him "Doctor Who," but it also said he was human so would it know, eh?
3a. Isn't it Theta Sigma?
Short answer: No. Both The Armageddon Factor and The Happiness Patrol establish it as a college nickname.
4. Who is Susan Foreman?
Short answer: If we take the TV!canon at face value, she's the Doctor's biological grandchild. According the Virgin NAs, she's actually the grandchild of some guy called the Other.
5. Where do baby Time Lords come from?
Short answer: TV!Canon is silent on the subject other than the general statement that Susan is the Doctor's granddaughter. According to the NAs, they're woven on Looms.
5a. Where do baby Doctors come from?
Short answer: Well, see, that one's tricky. The Doctor was apparently woven on a Loom (the NAs) and born to a human mother (the TVM) who could be called Penelope (the EDAs). And Creature From the Pit has him claiming to have been born under the sign of the Gallifrey maternity service. (But he might just have been joking.)
6. Are all Gallifreyans Time Lords?
Short answer: There's no canonical certainty on that one. There's bits here and there to suggest either answer. Non-TV!canon is generally more with the idea that there's a difference.
7. A whole bunch of stuff about regeneration.
Short answer: *handwaves*
8. What's the Time War? And what happened to Gallifrey?
Not very short answer: This one's complicated by the fact that the BBC books line (the EDAs) also had a huge war in which Gallifrey was destroyed. This is not the same Time War as the one the ninth Doctor's always going on about. The TV!canon Time War was fought between the Time Lords and the Daleks, and Gallifrey is no longer there as a result (quite what that means isn't entirely clear). Other species, like the Gelth, were affected by it. The Doctor believes himself to be the only surviving Time Lord. It's implied (in Dalek) that he's the one that ended it.
9. Various things about the TARDIS
Short answer: Old, not new, "borrowed," blue.
Well, that was all clear as mud, wasn't it?
(Corrections etc sought and so on.)