The short answer in no. Khan came to power in the 1990s (Space Seed) thus from that point on the Star Trek universe was different from our own.
The long answer is rather more complicated. In Enterprise, though I haven't seen the episode in a while, they had a story where the time line is reset. During the reset I seem to remember their was a scene with Bush and Blair shaking hands. However if Kharn existed then he would have been a ruler between 1992-1996. Obviously these two facts are in conflict with each other.
Basically I think the writer's forgot the history of the show. Enterprise also had an episode about augments - I'm not sure if Kharn was mentioned.
Furthermore in one of the Star Trek book series has a character called Domenica Corsi. She has an axe that was used by an ancestor in 2001. So the 9/11 might have happened but not in quite the way we know it.
There is currently no mention in any canon about it, but I don't see why not.
See this the star trek timeline is not based in reality. In the canon timeline mankind has global government and space travel ability by the 21 century. Lets not forget the augment/eugenics wars either.
As a matter of fact, footage of 9/11 appeared in the timestream sequence at the end of Stormfront pt II.
- and Stormfront pt II featured an altered timeline which was later restored to normal when Archer destroyed the temporal generator.
Okay, just because 9/11 was shown in ST:ENT "In A Mirror, Darkly" does not mean that the event did not also occur in the Prime universe. ST:ENT did not "stay" in the Mirror Universe subsequent to "In A Mirror, Darkly". That being said, one certainly wonders whether or not it was a "continuum event", as the Allied first-in-time development of nuclear weapons was.
And, what if Star Trek's writers posited that the general public knowledge of what is to us the "recent" completion of the project of mapping the human genome and revelations of genetic engineering of plant life and lower animals was delayed, and Humanity has in fact been messing around with this science for decades already, to the point of Human experimentation, but most simply did not know about it? Consider the uproar over the "ethical" usage of stem cell lines, already. Just because the timeline of these events are what they are in the U.S. or the Western world does not mean that elsewhere in the world they did not occur earlier. We only started becoming aware of a lot of Cold War-era internal events in the Soviet Union during the 1980s and 1990s.
And, the Eugenics Wars would have had to have an enemy. We know that events in Asia drastically diverged, but events in the Western world were not as drastically different in Star Trek. It seems to me that in ST:VOY "Future's End", in Rain Robinson's office, there was a model of a DY-100 class. In that timeline, at least, and probably in the "Prime", "corrected" timeline too, the DY-100 class S.S. Botany Bay did lift off from the Baikonur Cosmodrome in 1996 ProfessorTrek 23:55, July 30, 2011 (UTC)