1 The first time President Push was elected there was a problem with 'hanging chads'. Voters in some areas had to punch holes for their choice: Chad (paper) - Wikipedia, the free encyclopedia
It's a made-up name, not a person.
2 When Jack Straw, then the Foreign Secretary, went to the US, journalists called him 'Jack Shaw'; the idea is simply that they don't know his name properly.
3 Again, it's supoosed to show that the guy doesn't know much about the UK- he's mixing Trafalgar Square and Piccadilly Circus.
4 Fayed = the owner of Harrods (not in Trafalgar Square)
Student or Learner