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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.
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.
Rational numbers are closed under addition, subtraction, multiplication. They are not closed under division, since you can't divide by zero. However, rational numbers excluding the zero are closed under division.