Did Miles even mention he was going to meet him there? And wasn't Michael Caine wearing the exact same clothes in the airport that he was wearing in his earlier scenes?
When Cobb meets Caine at the university he gives Caine some presents for the kids for when he (Caine) goes to visit. Considering some time, even as long as a week, passes between the training of the team and setting things up it makes sense that Caine would arrive in America before Cobb.
I'm not sure he's wearing the exact same clothes but we can either chalk this up to them simply being the same style of clothes or simply the movie not wanting to spend money to buy two outfits for a character we see for all of five total minutes betweent he two end of the film.
As for how his name was cleared so fast, powerful movie CEO magic. Nothing more. Plenty of movies use this trick and raise the exact same question.