Yes.
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The quotient of two nonzero integers is the definition of a rational number. There are nonzero numbers other than integers (imaginary, rational non-integers) that the quotient of would not be a rational number. If the two nonzero numbers are rational themselves, then the quotient will be rational. (For example, 4 divided by 2 is 2: all of those numbers are rational).
The definition of a rational number is the quotient of any two nonzero integers.
Because that is how a rational number is defined!
Yes, always. That is the definition of a rational number.
Yes.