Yes, you can.
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The result will also be a rational number.
You can not add irrational numbers. You can round off irrational numbers and then add them but in the process of rounding off the numbers, you make them rational. Then the sum becomes rational.
Not necessarily. The sum of two irrational numbers can be rational or irrational.
Yes. Rational numbers are numbers that can be written as a fraction. Irrational numbers cannot be expressed as a fraction.
If a set is closed under an operation. then the answer will be a part of that set. If you add, subtract or multiply any two rational numbers you get another national number. But when it comes to division, it is closed except for one number and that is ZERO. eg 3.56 (rational number) ÷ 0 = no answer. Since no answer is not a rational number, that rational numbers are not closed under the operation of division.