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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 it is.
Yes. The empty set is closed under the two operations.
If you mean the set of non-negative integers ("whole numbers" is a bit ambiguous in this sense), it is closed under addition and multiplication. If you mean "integers", the set is closed under addition, subtraction, multiplication.