Yes.
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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.
Yes. When you add any whole numbers you get another whole number. That is what closed means in this context. The answer is still a whole number.
There is no counterexample because the set of whole numbers is closed under addition (and subtraction).
They form a closed set under addition, subtraction or multiplication.
Yes because being closed under an operation means that when the operation is performed on members of a set the result is also a member of the set, and when any two [members of the set of] whole numbers are added together the result of the addition is also a whole number which is, unsurprisingly, a member of the set of whole numbers.