Photocopiers are the opposite of thinking. What goes in, goes out, no transformation or creating of new data at all. Any change is just an accident, or an artifact of the technical process.
True, modern Photocopiers are not really Photocopiers any more and just a pc with scanner and printer. Those could be described as "thinking", in the same sense as any computer is thinking. But from my understanding, the original comment was explicitly about a pure copy-device, without any "brain" involved.
Not sure what you mean - there's a pattern of dots (the EURion constellation) on the banknotes of a number of countries that photocopiers recognize and will refuse to copy.
I could have been more descriptive, but yes, exactly what you described. The logic of: if i see this exact pattern, refuse to duplicate or make extremely apparent it's a fake.