No. It has a different additive inverses for each element.
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Yes.
Wrong! Not only is zero a real number, but it is the additive identity for the set of integers, rational numbers as well as real numbers.
Yes, it does.
There is only one set and it does have an additive identity.
It gives closure to the set of real numbers with regard to the binary operation of addition. This makes the set a ring. The additive inverse is used, sometimes implicitly, in subtraction.