S(N) is just S(N), it's not N+1 unless you define it that way. S(0) is not 1, it's S(0). The successor function doesn't really get mapped to numbers without an explicit additional mapping.
No, it's not. That's just the standard model of peano arithmetic. You can map the successor function to many different operations and objects that aren't even numbers.
Yes it is. "1" is just a label for what is also known as S(0) no mater what other ways you invent to refer to that element. Just like 一 is another label for the same concept.
