Kirk did say “Beam me up, Mr. Scott.” IMNSHO there’s a difference between a paraphrased quote, with a few words swapped around here and there (the Dirty Harry example in post #8 is another), and a quote invented from whole cloth, based maybe at most on one key word.
P.T. Barnum never said “There’s a sucker born every minute”.
Greta Garbo never said “I want to be alone.” What she said was “I want to be left alone”.
And in the animated episode, “The Infinite Vulcan,” (Yeah, I know. The animated series is not generally regarded as canon.) Kirk did say, “Kirk to Enterprise. Beam us up, scotty.”
Along these same lines, Vader’s famous line from the Empire Strikes Back- “No. I am your father.” is often misquoted as “Luke, I am your father.” Grr.
Sure she did: in Grand Hotel (first quote). I can confirm: they played the clip many times when she died.
AIUI there’s some question whether Isoruku Yamamoto ever said “I will run rampant in the Pacific for six months.” LIkewise, there’s some reason to question whether he said, “We have awoken the sleeping giant.” But I love the quotes anyways.
And nowhere in the stories did Watson ever said, “No shit, Sherlock!” 
The way I read it (in Modern Times, by Paul Johnson), it was “We can run wild for the first year, but I can say nothing of the second or the third year.”
A famous radio host dubbed “Uncle Don” was never fired for saying “That oughta hold the little bastards” without realizing his mike was still on.
Al Gore never stated that he was the sole inventor of the Internet- his actual quote was “During my service in the United States Congress, I took the initiative in creating the Internet.” The phraseology of this makes it sound like he was the only one who did anything about it, which is probably where the confusion came from.