So, for the purposes of answering the question in the title of this fair thread, there was a time when TAS was, for all intents and purposes, not a part of the official canon, primarily for legal reasons, even though GR was probably loathe to admit that he couldn't use a portion of his own creation, thus the "sour grapes" memo and statements to Richard Arnold and Susan Sackett. In typical Roddenberry style, though, he always made an effort to tell folks what they wanted to hear, so to Dorothy Fontana, David Gerrold, Bjo Trimble, et al, who had something to do with TAS (or a major reference work with TAS all over it), he never gave any indication that TAS was in any way taboo.
Once the legal issues were resolved (sadly, not until Roddenberry had passed), it was back in the official canon, and the references to TAS started flowing into DS9, and were flying pretty fast and furious in ENT.
So, yes, Virginia, TAS is canon.