Yes to all three.
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Yes -10 is an integer and a rational number
It is real, rational, integer and whole but not irrational nor counting.
No. In general, the square root of a positive integer is either a whole number, or an irrational number.
Yes. -10 = -10/1 which is of the form of one whole number over another whole number. Rational numbers have the form of one whole number over another whole number. Thus -10 is rational.
(*Is negative 10 an integer?) Yes, because an integer is a positive or negative whole number.