Yes, they are.
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Yes. The entire set of natural numbers is closed under addition (but not subtraction). So are the even numbers (but not the odd numbers), the multiples of 3, of 4, etc.
Quite simply, they are closed under addition. No "when".
No. A number cannot be closed under addition: only a set can be closed. The set of rational numbers is closed under addition.
Yes, the set is closed.
Yes.