In re: the Doctor, regenerations, and names.

So far, the new crew seems to be hewing to the old canon wherever possible; the books and audios don't seem to come into it yet, if ever. There've been strong hints in the expanded universe that the novels and audios take place in divergent timestreams, which seems to solve the problem quite neatly. There's a fair amount in the classic series about Time Lords and regenerations and whatnot that new fans don't always seem to be aware of, though.

Clearing firstly out of the way something wot annoys my friend David a lot: Time Lords get twelve regenerations. Twelve second chances, essentially. You've got to regenerate from something the first time, which means that, assuming no jiggery-pokery, we've got room for thirteen Doctors. It's used consistently this way throughout the classic series, and the math squares up in stuff like Mawdryn Undead where the Doctor talks about how many lives other people are trying to steal from him. I don't recall if it's ever come up on the show, but in the expanded universe the Doctor does quite clearly keep count and will politely identify himself by ordinal if someone asks -- particularly the Brigadier, who has met all of him by this point, and whom he's asked nicely to please not tell him too much about his own future.

The assumption of no jiggery-pokery is also a bad one. I've seen a lot of stuff on tumblr that purports that Time Lords are born with the ability to regenerate, which doesn't seem to be the case. There are a lot of Gallifreyans who aren't Time Lords; Leela stays there and marries one, in fact. Getting the title seems to involve a lot of schooling and jumping through stuffy bureaucratic hoops. References are consistently made to the Council of Time Lords granting regenerations, which suggests rather strongly that it's an ability integrated into Time Lord biology from the outside, and something they invented along with all the wibbly-wobbly timey-wimey stuff.

Various people have referred to stealing or granting regenerations. You're not supposed to be able to do it without being on Gallifrey and using the Matrix -- basically the Database of Rassilon; why they don't call it that is beyond me, since everything else on the damn planet seems to have been named for him -- although for obvious reasons this does not actually stop the really clever ones. The Council at one point offered to grant the Master more regenerations if he did them a favor, which ultimately turned out about how you suspect it did.

The Doctor has also managed a couple of half-incarnations, although I gather you're really not supposed to be able to do that. It seems to have to do with crossing, changing, or otherwise muddying his own timestream. There was a mysterious, indistinct figure referred to as "the Watcher" hanging around just before the Fourth Doctor regenerated into the Fifth, apparently to provide him with a bit of warning; the Watcher merged back into the Doctor as the regeneration began. Something about the process discombobulated him rather badly, and the Fifth Doctor spent his entire first serial trying to re-combobulate his brain chemistry whilst simultaneously dealing with Castrovalva.

The Sixth Doctor also ran into a much less benevolent future half-incarnation, the Valeyard. His identity was kept a mystery for most of the Trial of A Time Lord season, but it was eventually revealed that the Valeyard was the distillation of the Doctor's evil side from "between his twelfth and final incarnations". Please don't ask me how that works; I don't know. Wibbly-wobbly timey-wimey. The Doctor also did some stirring around in his own timeline during that, although it was more hands-off than normal -- one of the incidents which got him hauled up on charges was one from the future of the Doctor on trial, with a companion he hadn't met yet, and whose origins we never see. (Big Finish has just now gotten around to explaining how he picked up Mel in one of the audios, in fact.)

As for the Doctor's real name, it really has sincerely never been touched upon until now. "The Doctor" is the only way he's ever introduced himself, except for when he was with UNIT and they demanded more than that for their official paperwork. He spends several minutes being as resentful and obtuse as he possibly can. The poor typist eventually drags "Dr. John Smith" out of him, and decides to believe that just to make it stop. It's like pulling teeth. He uses the alias pretty consistently from then on out, except for Ten's brief stopover in Scotland with Rose, where he gives "Jamie McCrimmon", the name of a former companion who was from the 18th c. Scottish highlands.

The other Time Lords consistently call him "Doctor" when he gets hauled back to Gallifrey, even when they aren't asking him for absurd favors, except for one old classmate who sees fit to remind him that his nickname in school was "Theta Sigma". The Doctor seems to find this vaguely embarrassing, for reasons that aren't explained. Other Time Lords have lengthy names which are implied to carry some sort of information about lineage or credentials, of which there are also contracted versions for casual use. "Romana" is short for "Romanadevoratrelundar," evidently, and in the expanded universe there are other Time Lords and Ladies with hideously tongue-twisting names that get used exactly once before everyone reverts to the short form.

The written Time Lord language is unique to the new series. Languages, I should say -- the circular stuff looks mostly like graphic design to me and doesn't whack the language sensors, but the ancient scratchings in River's 'hello, sweetie' message look plausible enough, at least with a small sample like that. The one appearance of any of this in the classic show was some really ancient runic things in The Five Doctors, but these are so old that they are no longer everyday language there, and it takes multiple bickering Doctors to even guess at what they're meant to say.

The TARDIS has a quirky sense of humor about, "Hello, I'm the Doctor." The Doctor does seem to legitimately speak a number of natural languages, but the rest are handled by the TARDIS' telepathic translation circuits. Whatever word he's actually using as his name, she seems to have a fondness for rendering it whenever possible into something that preserves the collision between the "highly-educated person" and "medical professional" senses. He gets mistaken for the local equivalent of a healer or an MD quite a lot, strictly on grounds of verbal ambiguity. Given his fondness for England, he might actually literally be using the word "Doctor" -- the problem of crossing meanings isn't even present in all Earth languages, but it is in most of Europe, where it descends largely from historical quirks of the university system.