1.What does "accident of timing" mean here?
It means that it was not just an accident or coincidence that she wore a youthful hairstyle to this ceremony and a stylish dress to the same ceremony four years ago. The implication is that she was making a particular point by the way she presented herself.
2.Who are "they"? What does "it" refer to, and what outfit is "that outfit"?
"they" = the President and his wife
"that outfit" = the dress she was wearing
"what it said" = the point that was being made, which, according to the writer, was "the nation is changing".
3.Was the President saying that the first lady's haircut was the most significant event of this weekend?
Because the haircut received so much media attention, the President was jokingly suggesting it was the most significant event.
Note the corrections I've made to your questions.
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