Pulling names out of a hat can easily be manipulated and staged.
No it can't. It's fraud and production companies don't risk huge fines/having a channels broadcasting license revoked since the fakery issues of the mid 00s (actually pretty minor; the most serious were on viewer phone-in competitions and one case of a viewer song selection vote on 'Soapstar Superstat' being disregarded).
In terms of mainstream media, TV is actually the most compliance heavy by far. If you run a competition or say something is random, the actual mechanics are it are whiter than white. The producers can influence the *opinion* of the public on phone vote through editing, running orders etc - but they never lie about the results of a phone vote or competition. It just doesn't happen.
And I think Cowell is saying in that interview that the running order will be more 'mixed up' and less predictable - not that it will actually be randomised. If it isn't genuinely random they will NEVER refer to randomness in the wording of the script or press releases.