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Yes.
Rational numbers are closed under multiplication, because if you multiply any rational number you will get a pattern. Rational numbers also have a pattern or terminatge, which is good to keep in mind.
Yes. They are closed under addition, subtraction, multiplication. The rational numbers WITHOUT ZERO are closed under division.
The set of whole numbers is not closed under division (by non-zero whole numbers).
No. To say a set is closed under multiplication means that if you multiply any two numbers in the set, the result is always a member of the set. If, say, the 2 numbers are radical 2 and radical 2 we have (1.4142...)(1.4142...) which by definition equals 2. The result is not an irrational number, so the set is not closed.