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What are the marks on some stamps (such as on the American ones (the curved lines) that I uploaded below, versus the German ones that don't have them. And how does this impact the stamp's value?)
As a general rule, mint (unused) stamps are worth more than cancelled (used) stamps. There are some exceptions, such as when a relatively rare stamp was in actual postal use for a very short time. In those cases, the cancellation needs to be from the proper period of use (and the stamp is preferably still on the envelope or postcard) and usually needs to be authenticated. Which brings me to another one of your questions:
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How to tell if a stamp is genuine?
Short answer: Compare it with one that is known to be genuine.
There is no easy answer to this question. Some fakes are obvious, others less so. Some "classic" fakes are even worth something. Being able to tell the difference between genuine and fake often requires study, especially when the stamp is genuine and the overprint or cancellation on it is fake.