For example: There is nothing to say that at least the old Dr. Who series did not exist in Trek universe (the new show would not have existed since its creation would fall right into WWIII if I´m not mistaken).
Say what? You must never have seen one of the classic Doctor Who episodes: "The Dalek Invasion of Earth" - the First Doctor, with companions Susan, Ian, and Barbara. It states that in 2164, the Daleks had taken over Earth, and the humans still alive were either slaves to the Daleks, or rebels trying to overthrow them. I don't recall 22nd-century Earth in the Star Trek universe being under the control of cybernetic aliens from the planet Skaro. And some of the Jon Pertwee-era UNIT material was set later in the 20th century than the 1970s, since there were time-traveling Daleks involved.