Every integer is a rational number, and some integers are perfect squares. These are the only rational numbers to have an integral square root.
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No. Square roots of whole numbers that don't result in whole numbers (e.g. the square root of 64 is 8), are considered to be "Irrational Numbers".
Yes. Not only that, they are counting numbers.
Very rarely. Square roots are only whole numbers for perfect squares.
Some square roots of whole numbers are integers, some of them are irrational numbers. The square root of four for instance is a rational number, 2. The square root of two however is an irrational number, approximately 1.414211356.
Whole numbers ARE rational.