Another thing you have to watch out for is when canon incorporates the statements of a character who is an unreliable narrator or who lies. In some cases, fanon claims that the original narrator is unreliable (often with a good excuse, such as changing names and situations to protect the innocent,) in order to clear up inconsistencies in canon. This seems to happen a lot with Sherlockians, in order to clear up such questions as which cases happened first, how many times was Watson married, and why anyone thought that snakes could digest milk. (I'm not even going to get into The Creeping Man. I think for most Sherlockians who were unfortunate enough to read that after starting biology it was a big WTF moment, not that we can blame ACD for not knowing better any more than we can blame Jules Verne for not knowing that Nemo and company would be instantly crushed if they ventured outside of the Nautilus in the middle of the Atlantic Ocean.)

The only series I have any experience of with officially sanctioned fanfiction is the Star Wars Expanded Universe, which seems to me to be a vast collection of Alternate Universes held together by the application of retcons, duct tape, and Timothy Zahn. Unless you were writing fanfiction of a specific EU book or with a specific EU character, I doubt you'd have to bring in any EU "canon" elements.

The Last Airbender - I can only presume that you refer to the TV series that proceeded The Legend of Korra, so yes, it has to be canon, doesn't it? ;)

I think I would have noticed if there was a movie for Avatar: The Last Airbender. It's too bad there hasn't been! I'm sure it would be a great film, in the right hands.

I'm sort of a mild, toe-in-the-water, Doctor Who fan, so don't place too much importance on my answer.
As Neshomeh mentioned, quite a few canons do have hierarchies, so although I don't know of an established one for Doctor Who, I'd say that the shows would come first for canonical resources, and then the novels. With DW, though, it's hard to be completely certain.
On a completely irrelevant sidetrack: I liked some of the Last Airbender. The scenery was quite wonderful in parts, but the acting was highly amateur, and most of the time Aang was completely out of context for emotional reactions. Or it seemed fake. And they pronounced his name wrong.
Anyway, hope this helped!

I know some canons that will accept anything that doesn't directly contradict established material, and throws the entire installment it out the window when established events are broached, and others that officially license Expanded Universe works and treat them as just as canonical as the main source. Then there are others that accept any material given to them, but throw certain volumes out selectively for being stupid, driving people out of character, etcetera. Then there are universes like Star Trek, in which the episodes are the only canon, and all of the books and comics are shunted off into alternate universes that have become so tightly knit with each other's backstories and histories that they might serve as an independent canon if necessary.

Just because someone has a license to write in the universe, that doesn't mean that the creators want the material, or that they will accept it. Even if official policy makes a certain material canonical, an unaware or spiteful writer might deliberately write over it, figuratively speaking, if it doesn't mesh with his views of how the story should go, even if some of the written-over material was previously part of the "alpha" canon, which creates what is commonly called Continuity Snarl.

In answer to your two direct questions, yes, the tie-in books of most franchises are officially sanctioned fanfiction. Even the Star Wars Expanded Universe, which makes up almost all of the Star Wars material, is technically fanfiction, and the people at Lucasfilm can and will steamroll over it if they think they have a good idea for a new video game or some such.
An EU can be treated as canon, but only the central portions of the franchise are "real" in most cases, and sometimes, not even then. In continuities prone to mass retconning, like DC Comics, this latter point can be a big problem.

Doctor Who EU in particular was, at least up to the end of the Russel T. Davies era, treated as canonical if it didn't try to mess with the established material of episodes to make a story with "higher stakes" or some such. See War of the Daleks for a particularly blatant example of attempted retconning that was treated as, as I've heard said "a big no-no".
I don't know how the Steven Moffat era treats it. I've seen one episode where the Doctor mentions a few species only ever seen in the EU, and I've seen plenty more where several previously established points in Classic Who and the EU were abandoned for cheap kicks.

For your last question, the M. Night Shyamalan The Last Airbender movie is not considered canon to the cartoon. Both the fans and creators have disavowed it, and at any rate, since it takes place within the same timeframe as canonical events but treats them far differently, to say nothing about the OOC that we see rampant in the storyline, it would be an Alternate Universe story at best.

(By the way, what were the inconsistencies specific to Asylum of the Daleks? I watch Doctor Who on Netflix, so I've not seen most of Series Seven yet.)

Some of them have sort of a canon hierarchy—for instance, I know Star Wars has a ranking system for canon materials, with the six films and Word of God at the top. Lower-ranked materials tend to be considered canon only if they don't contradict the materials ranked above them. Same goes for the Warcraft universe: World of Warcraft is at the top, followed by previous games, followed by books and other stuff. Using elements from lower-ranked materials in a fic supposedly set in the "alpha" canon can be problematic, but if it's justified and works well in the story, there's not much to complain about. Doctor Who probably works like this, but someone more familiar with the fandom can tell you better than I can.

Others, like Harry Potter and The Lord of the Rings, have separate movieverse and bookverse (and gameverse and whatever-elseverse) canons. They're protected equally by the PPC, but are distinct from each other, and trying to use movieverse elements in an explicitly bookverse fic (or vice versa) can be problematic and sometimes charge-worthy. I think the A:TLA movieverse vs. showverse fits into this category.

I hope that helps!

~Neshomeh