What is "Canon"?
Canon is typically defined as any movie or television / streaming production of a given franchise, not including the subcategories of canon that follow.
Alpha Canon includes officially licensed print materials, such as movie tie-in comics, novelizations of canon productions, and the like.
Beta Canon includes productions licensed but not produced by the owner of a given franchise, such as novels and other published works of fiction and non-fiction. Note, that Star Wars novels were generally accepted as Canon until the Disney acquisition of LucasFilm.
For the sake of this guide, only official canon material is acknowledged.
When canon is found to contradict itself, we find it best to assume the newest information to be official as long as it can be justified in universe. Otherwise, assume it is correct for that series / movie.
DC has released concurrent entries into their franchise, often considering conflicting / seemingly non-canon stories as Elseworlds timelines. This is also explored in The Flash, for example.
While not canon, "Star Trek Continues" has been included in our Star Trek lists. The fan-made series is true to the spirit of Roddenberry's vision and delivers quality writing, acting, and film editing.
Is the Millennium Falcon in the Star Trek canonical universe?
Yes. In "Star Trek: First Contact", Starfleet received some assistance from the ship that could make the Kessel Run in fourteen twelve parsecs.
Reference: This article at Eggpedia
The Official Star Trek Cannon fires Star Trek DVDs and is wholly ineffective as a siege weapon.